Abstract

A generalized exponential spectrum model is derived, which considers finite turbulence inner and outer scales and has a general spectral power law value between the range 3 to 5 instead of standard power law value 11/3. Based on this generalized spectrum model, a new generalized long exposure turbulence modulation transfer function (MTF) is obtained for optical plane and spherical wave propagating through horizontal path in weak fluctuation turbulence. When the inner scale and outer scale are set to zero and infinite, respectively, the new generalized MTF is reduced to the classical generalized MTF derived from the non-Kolmogorov spectrum.

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References

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  1. V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation, (trans.for NOAA by Israel Program for Scientific Translations, Jerusalem, 1971).
  2. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media. (SPIE Optical Engineering Press, Bellingham, 2005).
  3. D. T. Kyrazis, J. Wissler, D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2120, 43–55 (1994).
    [CrossRef]
  4. M. S. Belen’kii, S. J. Karis, J. M. Brown, and R. Q. Fugate, “Experimental study of the effect of non-Kolmogorov stratospheric turbulence on star image motion,” Proc. SPIE 3126, 113–123 (1997).
    [CrossRef]
  5. 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, 63040U (2006).
    [CrossRef]
  6. A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov trubulence,” Atmos. Res. 88(1), 66–77 (2008).
    [CrossRef]
  7. A. Zilberman, and N. S. Kopeika, “Slant-path generalized atmospheric MTF,” in Proceedings of IEEE 25th Convention of Electrical and Electronics Engineers (Institute of Electrical and Electronics Engineers, Israel, 2008), pp. 217–221.
  8. N. S. Kopeika, A. Zilberman, and E. Golbraikh, “Generalized atmospheric turbulence: implications regarding imaging and communications,” Proc. SPIE 7588, 758808 (2010).
    [CrossRef]
  9. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of Arrival Fluctuations for Free Space Laser Beam Propagation through non Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
    [CrossRef]
  10. L. C. Andrews, Special Functions of Mathematics for Engineers, 2nd ed. (SPIE Optical Engineering Press, Bellingham, Wash., 1998).
  11. R. E. Hufnagel and N. R. Stanley, “Modulation Transfer Function associated with Image Transmission through Turbulent Media,” J. Opt. Soc. Am. 54(1), 52–61 (1964).
    [CrossRef]
  12. D. L. Fried, “Optical Resolution Through a Randomly Inhomogeneous Medium for Very Long and very Short Exposures,” J. Opt. Soc. Am. 56(10), 1372–1379 (1966).
    [CrossRef]
  13. B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical Propagation in non-Kolmogorov Atmospheric Turbulence,” Proc. SPIE 2471, 181–196 (1995).
    [CrossRef]
  14. 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(34), 6385–6391 (2008).
    [CrossRef] [PubMed]
  15. W. B. Miller, J. C. Ricklin, and L. C. Andrews, “Log-amplitude variance and wave structure function: a new perspective for Gaussian beams,” J. Opt. Soc. Am. A 10(4), 661–672 (1993).
    [CrossRef]

2010

N. S. Kopeika, A. Zilberman, and E. Golbraikh, “Generalized atmospheric turbulence: implications regarding imaging and communications,” Proc. SPIE 7588, 758808 (2010).
[CrossRef]

2008

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov trubulence,” Atmos. Res. 88(1), 66–77 (2008).
[CrossRef]

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(34), 6385–6391 (2008).
[CrossRef] [PubMed]

2007

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of Arrival Fluctuations for Free Space Laser Beam Propagation through non Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[CrossRef]

2006

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, 63040U (2006).
[CrossRef]

1997

M. S. Belen’kii, S. J. Karis, J. M. Brown, and R. Q. Fugate, “Experimental study of the effect of non-Kolmogorov stratospheric turbulence on star image motion,” Proc. SPIE 3126, 113–123 (1997).
[CrossRef]

1995

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

1994

D. T. Kyrazis, J. Wissler, D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2120, 43–55 (1994).
[CrossRef]

1993

1966

1964

Andrews, L. C.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of Arrival Fluctuations for Free Space Laser Beam Propagation through non Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[CrossRef]

W. B. Miller, J. C. Ricklin, and L. C. Andrews, “Log-amplitude variance and wave structure function: a new perspective for Gaussian beams,” J. Opt. Soc. Am. A 10(4), 661–672 (1993).
[CrossRef]

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, 63040U (2006).
[CrossRef]

M. S. Belen’kii, S. J. Karis, J. M. Brown, and R. Q. Fugate, “Experimental study of the effect of non-Kolmogorov stratospheric turbulence on star image motion,” Proc. SPIE 3126, 113–123 (1997).
[CrossRef]

Bishop, K. P.

D. T. Kyrazis, J. Wissler, D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2120, 43–55 (1994).
[CrossRef]

Brown, J. M.

M. S. Belen’kii, S. J. Karis, J. M. Brown, and R. Q. Fugate, “Experimental study of the effect of non-Kolmogorov stratospheric turbulence on star image motion,” Proc. SPIE 3126, 113–123 (1997).
[CrossRef]

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, 63040U (2006).
[CrossRef]

Ferrero, V.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of Arrival Fluctuations for Free Space Laser Beam Propagation through non Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[CrossRef]

Fried, D. L.

Fugate, R. Q.

M. S. Belen’kii, S. J. Karis, J. M. Brown, and R. Q. Fugate, “Experimental study of the effect of non-Kolmogorov stratospheric turbulence on star image motion,” Proc. SPIE 3126, 113–123 (1997).
[CrossRef]

Golbraikh, E.

N. S. Kopeika, A. Zilberman, and E. Golbraikh, “Generalized atmospheric turbulence: implications regarding imaging and communications,” Proc. SPIE 7588, 758808 (2010).
[CrossRef]

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(34), 6385–6391 (2008).
[CrossRef] [PubMed]

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov trubulence,” Atmos. Res. 88(1), 66–77 (2008).
[CrossRef]

Hufnagel, R. E.

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, 63040U (2006).
[CrossRef]

Karis, S. J.

M. S. Belen’kii, S. J. Karis, J. M. Brown, and R. Q. Fugate, “Experimental study of the effect of non-Kolmogorov stratospheric turbulence on star image motion,” Proc. SPIE 3126, 113–123 (1997).
[CrossRef]

Keating, D. B.

D. T. Kyrazis, J. Wissler, D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2120, 43–55 (1994).
[CrossRef]

Kopeika, N. S.

N. S. Kopeika, A. Zilberman, and E. Golbraikh, “Generalized atmospheric turbulence: implications regarding imaging and communications,” Proc. SPIE 7588, 758808 (2010).
[CrossRef]

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(34), 6385–6391 (2008).
[CrossRef] [PubMed]

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov trubulence,” Atmos. Res. 88(1), 66–77 (2008).
[CrossRef]

Kupershmidt, I.

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov trubulence,” Atmos. Res. 88(1), 66–77 (2008).
[CrossRef]

Kyrazis, D. T.

D. T. Kyrazis, J. Wissler, D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2120, 43–55 (1994).
[CrossRef]

Miller, W. B.

Phillips, R. L.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of Arrival Fluctuations for Free Space Laser Beam Propagation through non Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[CrossRef]

Preble, A. J.

D. T. Kyrazis, J. Wissler, D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2120, 43–55 (1994).
[CrossRef]

Ricklin, J. C.

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, 63040U (2006).
[CrossRef]

Shtemler, Y.

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov trubulence,” Atmos. Res. 88(1), 66–77 (2008).
[CrossRef]

Stanley, N. R.

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, “Angle of Arrival Fluctuations for Free Space Laser Beam Propagation through non Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[CrossRef]

Virtser, A.

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov trubulence,” Atmos. Res. 88(1), 66–77 (2008).
[CrossRef]

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]

Wissler, J.

D. T. Kyrazis, J. Wissler, D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2120, 43–55 (1994).
[CrossRef]

Zilberman, A.

N. S. Kopeika, A. Zilberman, and E. Golbraikh, “Generalized atmospheric turbulence: implications regarding imaging and communications,” Proc. SPIE 7588, 758808 (2010).
[CrossRef]

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(34), 6385–6391 (2008).
[CrossRef] [PubMed]

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov trubulence,” Atmos. Res. 88(1), 66–77 (2008).
[CrossRef]

Appl. Opt.

Atmos. Res.

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov trubulence,” Atmos. Res. 88(1), 66–77 (2008).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Proc. SPIE

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

N. S. Kopeika, A. Zilberman, and E. Golbraikh, “Generalized atmospheric turbulence: implications regarding imaging and communications,” Proc. SPIE 7588, 758808 (2010).
[CrossRef]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of Arrival Fluctuations for Free Space Laser Beam Propagation through non Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[CrossRef]

D. T. Kyrazis, J. Wissler, D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2120, 43–55 (1994).
[CrossRef]

M. S. Belen’kii, S. J. Karis, J. M. Brown, and R. Q. Fugate, “Experimental study of the effect of non-Kolmogorov stratospheric turbulence on star image motion,” Proc. SPIE 3126, 113–123 (1997).
[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, 63040U (2006).
[CrossRef]

Other

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation, (trans.for NOAA by Israel Program for Scientific Translations, Jerusalem, 1971).

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media. (SPIE Optical Engineering Press, Bellingham, 2005).

L. C. Andrews, Special Functions of Mathematics for Engineers, 2nd ed. (SPIE Optical Engineering Press, Bellingham, Wash., 1998).

A. Zilberman, and N. S. Kopeika, “Slant-path generalized atmospheric MTF,” in Proceedings of IEEE 25th Convention of Electrical and Electronics Engineers (Institute of Electrical and Electronics Engineers, Israel, 2008), pp. 217–221.

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Figures (4)

Fig. 1
Fig. 1

A ( α ) and c ( α ) as a function of α. (a): A ( α ) , (b): c ( α )

Fig. 2
Fig. 2

MTF as a function of normalized spatial frequency with different outer scale values. (a): plane wave. (b): spherical wave.

Fig. 3
Fig. 3

MTF as a function of normalized spatial frequency with different inner scale values. (a): plane wave. (b): spherical wave

Fig. 4
Fig. 4

MTF as a function of normalized spatial frequency with different α values. (a): plane wave. (b): spherical wave

Equations (55)

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Φ n ( κ ) = 0.033 C n 2 κ 11 / 3 exp ( κ 2 κ l 2 ) [ 1 exp ( κ 2 κ 0 2 ) ] ( 0 k < )
Φ n ( κ , α ) = A ( α ) C n 2 ^ F ( k , l 0 , L 0 , α ) ( 0 κ < , 3 < α < 5 )
Φ n ( κ , α , l 0 , L 0 ) = A ^ ( α ) C n 2 ^ κ α f ( k , l 0 , L 0 , α ) ( 0 κ < , 3 < α < 5 )
f ( κ , l 0 , L 0 , α ) = exp ( κ 2 κ l 2 ) [ 1 exp ( κ 2 κ 0 2 ) ]
D n ( R ) = 8 π 0 κ 2 Φ n ( k ) ( 1 sin κ R κ R ) d κ
D n ( R , α ) = 8 π 0 κ 2 Φ n ( k , α ) ( 1 sin κ R κ R ) d κ
D n ( R , α ) = 8 π 0 κ 2 α A ^ ( α ) C n 2 ^ exp ( κ 2 κ l 2 ) ( 1 sin κ R κ R ) d κ
1 sin κ R κ R = n = 1 ( 1 ) n 1 ( 2 n + 1 ) ! κ 2 n R 2 n
D n 1 ( R , α ) = 8 π A ^ ( α ) C n 2 ^ n = 1 ( 1 ) n 1 ( 2 n + 1 ) ! R 2 n 0 κ 2 α + 2 n exp ( κ 2 κ l 2 ) d κ
Γ ( x ) = 0 κ x 1 e κ d κ ( κ > 0 , x > 0 ) , F 1 1 ( a ; b ; z ) = n = 0 ( a ) n z n ( b ) n n !
( a ) n = Γ ( a + n ) Γ ( a ) = a ( a + 1 ) ( a + n 1 )
D n ( R , α ) = 4 π A ^ ( α ) C n 2 ^ κ l 3 α { Γ ( α 2 + 3 2 ) [ 1 F 1 1 ( α 2 + 3 2 ; 3 2 ; R 2 κ l 2 4 ) ] }
D n ( R , α ) = { C n 2 ^ l 0 α 5 R 2 0 R l 0 C n 2 ^ R α 3 l 0 R L 0
F 1 1 ( a ; b ; x ) Γ ( b ) Γ ( b a ) x a ( x 1 )
D n ( R , α ) 4 π A ^ ( α ) C n 2 ^ Γ ( α 2 + 3 2 ) Γ ( 3 2 ) Γ ( α 2 ) ( 1 2 ) α 3 ( R ) α 3 ( l 0 R L 0 )
Γ ( α + 1 ) = α Γ ( α )
Γ ( 1 α ) Γ ( α ) = π sin ( π α )
Γ ( α ) Γ ( α + 1 2 ) = 2 1 2 α π Γ ( 2 α )
A ^ ( α ) = Γ ( α 1 ) 4 π 2 sin [ ( α 3 ) π 2 ]
F 1 1 ( a ; b ; x ) n = 0 1 ( a ) n z n ( b ) n n ! = 1 + a b x ( x 1 )
D n ( R , α ) π A ^ ( α ) C n 2 ^ κ l 5 α R 2 [ Γ ( α 2 + 3 2 ) ( 3 α 3 ) ] ( 0 R l 0 )
c ( α ) = { π A ^ ( α ) [ Γ ( α 2 + 3 2 ) ( 3 α 3 ) ] } 1 α 5
Φ n ( κ , α ) = A ^ ( α ) C n 2 ^ κ α ( 0 κ < , 3 < α < 5 )
M T F t u r b ( ν ) = exp [ 1 2 D ω ( λ F ν ) ]
D ω p ( ρ ) = 8 π 2 k 2 0 L d z 0 [ 1 J 0 ( κ ρ ) ] Φ n ( κ , z ) κ d κ
D ω s ( ρ ) = 8 π 2 k 2 0 L d z 0 [ 1 J 0 ( κ ρ z / L ) ] Φ n ( κ , z ) κ d κ
D ω p ( ρ , α , l 0 , L 0 ) = 8 π 2 k 2 0 L d z 0 [ 1 J 0 ( κ ρ ) ] Φ n ( κ , α , l 0 , L 0 ) κ d κ ( 3 < α < 4 )
J 0 ( x ) = n = 0 ( 1 ) n n ! Γ ( n + 1 ) ( x 2 ) 2 n
D ω p ( ρ , α , l 0 , L 0 ) = 8 π 2 k 2 A ^ ( α ) C n 2 ^ 0 L { n = 1 ( 1 ) n 1 n ! ( 1 ) n ( ρ 2 ) 2 n 0 κ 2 n α + 1 f ( κ , l 0 , L 0 , α ) d κ } d z
D ω p ( ρ , α , l 0 , L 0 ) = 8 π 2 k 2 A ^ ( α ) C n 2 ^ L n = 1 ( 1 ) n 1 n ! ( 1 ) n ( ρ 2 4 ) n 0 κ 2 n α + 1 f ( κ , l 0 , L 0 , α ) d κ
f 1 ( κ , l 0 , L 0 , α ) = exp ( κ 2 κ l 2 )
f 2 ( κ , l 0 , L 0 , α ) = exp [ κ 2 ( 1 κ 0 2 + 1 κ l 2 ) ]
D ω p ( ρ , α , l 0 , L 0 ) = D ω p 1 ( ρ , α , l 0 , L 0 ) + D ω p 2 ( ρ , α , l 0 , L 0 )
D ω p 1 ( ρ , α , l 0 , L 0 ) = 8 π 2 k 2 A ^ ( α ) C n 2 ^ L κ l 2 α { 1 2 Γ ( α 2 + 1 ) [ 1 F 1 1 ( α 2 + 1 ; 1 ; ρ 2 κ l 2 4 ) ] }
D ω p 2 ( ρ , α , l 0 , L 0 ) = 8 π 2 k 2 A ^ ( α ) C n 2 ^ L { 1 2 κ 2 α 2 1 Γ ( α 2 + 1 ) [ 1 F 1 1 ( α 2 + 1 ; 1 ; ρ 2 4 κ 2 ) ] }
D ω p 1 ( ρ , α , l 0 , L 0 ) 2 4 α π 2 k 2 A ^ ( α ) C n 2 ^ L Γ ( α / 2 + 1 ) Γ ( α / 2 ) ρ α 2 ( 3 < α < 4 )
D ω p 2 ( ρ , α , l 0 , L 0 ) 0 ( 3 < α < 4 )
D ω s ( ρ , α , l 0 , L 0 ) = 8 π 2 k 2 0 L d z 0 [ 1 J 0 ( κ ρ z / L ) ] Φ n ( κ , α , l 0 , L 0 ) κ d κ ( 3 < α < 4 )
D ω s ( ρ , α , l 0 , L 0 ) = 8 π 2 k 2 A ^ ( α ) C n 2 ^ 0 L { n = 1 ( 1 ) n 1 n ! ( 1 ) n ( ρ z 2 L ) 2 n 0 κ 2 n α + 1 f ( κ , l 0 , L 0 , α ) d κ } d z
D ω s ( ρ , α , l 0 , L 0 ) = 8 π 2 k 2 A ^ ( α ) C n 2 ^ L n = 1 { ( 1 ) n 1 n ! ( 1 ) n 1 2 n + 1 ( ρ 2 ) 2 n 0 κ 2 n α + 1 f ( κ , l 0 , L 0 , α ) d κ }
F 2 2 ( a , b ; c , d ; z ) = n = 0 ( a ) n ( c ) n z n ( b ) n ( d ) n n !
D ω s ( ρ , α , l 0 , L 0 ) = D ω s 1 ( ρ , α , l 0 , L 0 ) + D ω s 2 ( ρ , α , l 0 , L 0 )
D ω s 1 ( ρ , α , l 0 , L 0 ) = 8 π 2 k 2 A ^ ( α ) C n 2 ^ L κ l 2 α { 1 2 Γ ( α 2 + 1 ) [ 1 F 2 2 ( α 2 + 1 , 1 2 ; 1 , 3 2 ; ρ 2 κ l 2 4 ) ] }
D ω s 2 ( ρ , α , l 0 , L 0 ) = 8 π 2 k 2 A ^ ( α ) C n 2 ^ L { 1 2 κ 2 α 2 1 Γ ( α 2 + 1 ) [ 1 F 2 2 ( α 2 + 1 , 1 2 ; 1 , 3 2 ; ρ 2 4 κ 2 ) ] }
F 2 2 ( a , b ; c , d ; x ) Γ ( c ) Γ ( d ) Γ ( b a ) Γ ( b ) Γ ( c a ) Γ ( d a ) x a + Γ ( c ) Γ ( d ) Γ ( a b ) Γ ( a ) Γ ( c b ) Γ ( d b ) x b ( x 1 )
F 2 2 ( a , b ; c , d ; z ) n = 0 1 ( a ) n ( c ) n z n ( b ) n ( d ) n n ! = 1 + a b c d x ( x 1 )
D ω s 1 ( ρ , α , l 0 , L 0 ) 1 α 1 ( 2 4 α π 2 k 2 A ^ ( α ) C n 2 ^ L Γ ( α / 2 + 1 ) Γ ( α / 2 ) ρ α 2 ) ( 3 < α < 4 )
D ω s 2 ( ρ , α , l 0 , L 0 ) 0 ( 3 < α < 4 )
M T F t u r b ( p l ) ( u , α , l 0 , L 0 ) = exp [ 1 2 D ω p ( u D , α , l 0 , L 0 ) ] ( 3 < α < 4 )
M T F t u r b ( s p ) ( u , α , l 0 , L 0 ) = exp [ 1 2 D ω s ( u D , α , l 0 , L 0 ) ] ( 3 < α < 4 )
M T F t o t a l ( u ) = 2 π [ cos 1 u u 1 u 2 ] × M T F t u r b ( u ) × M T F a e r s o l ( u )
C n 2 ^ = 1.6 × 10 14 m 3 α
λ = 1.55 μ m
L = 1000 m
D = 0.1 m

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