Abstract

In the past decades, both the increasing experimental evidences and some results of theoretical investigation on non-Kolmogorov turbulence have been reported. This has prompted the study of optical propagation in non-Kolmogorov atmospheric turbulence. In this paper, using a non-Kolmogorov power spectrum which owns a generalized power law instead of standard Kolmogorov power law value 11/3 and a generalized amplitude factor instead of constant value 0.033, the log-amplitude variances for a Gaussian-beam wave are derived in the weak-fluctuation regime for a horizonal path. The analytic expressions are obtained and then used to analyze the effect of spectral power-law variations on the log-amplitude fluctuations of Gaussian-beam wave.

© 2010 Optical Society of America

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References

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  1. M. S. Belen’kii, S. J. Karis, J. M. Brown, and R. Q. Fugate, "Experimental evidence of the effects of non-Kolmogorov turbulence and anisotropy of turbulence," Proc. SPIE 3749, 50-51 (1999).
    [CrossRef]
  2. 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]
  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 2110, 43-55 (1994).
    [CrossRef]
  4. G. Wang, "A new random-phase-screen time series simulation algorithm for dynamically atmospheric turbulence wave-front generator," Proc. SPIE 6027, 602716-1-12 (2006).
  5. 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 turbulence," Atmospheric Research 88, 66-77 (2008).
    [CrossRef]
  6. A. Zilberman, E. Golbraikh, and N. S. Kopeika, "Lidar studies of aerosols and non-Kolmogorov turbulence in the Mediterranean troposphere," Proc. SPIE 5987, 598702-1-12 (2005).
  7. G. K. Batchelor, "Small-scale variation of convected quantities like temperature in turbulent fluid. Part I. General discussion and the case of small conductivity," J. Fluid Mech. 5, 113-133 (1959).
    [CrossRef]
  8. E. Golbraikh and N. S. Kopeika, "Behavior of structure function of refraction coefficients in different trubulent fields," Appl. Opt. 43, 6151-6156 (2004).
    [CrossRef] [PubMed]
  9. S. S. Moiseev and O. G. Chkhetiani, "Helical scaling in turbulence," JETP 83, 192-198 (1996).
  10. T. Elperin, N. Kleeorin, and I. Rogachevskii, "Isotropic and anisotropic spectra of passive scalar fluctuations in turbulent fluid flow," Phys. Rev. E 53, 3431-3441 (1996).
    [CrossRef]
  11. R. R. Beland, "Some aspects of propagation through weak isotropic non-Kolmogorov turbulence," Proc. SPIE 2375, 1111-1126 (1995).
  12. B. E. Stribling, B. M. Welsh, and M. C. Roggemann, "Optical propagation in non-Kolmogorov atmospheric turbulence," Proc. SPIE 2471, 181-196 (1995).
    [CrossRef]
  13. G. D. Boreman and C. Dainty, "Zernike expansions for non-Kolmogorov turbulence," J. Opt. Soc. Am. A 13, 517-522 (1996).
    [CrossRef]
  14. C. Rao, W. Jiang, and N. Ling, "Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence," J. Mod. Opt. 47, 175-177 (2000).
    [CrossRef]
  15. 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]
  16. M. S. Belen’kii, "Effect of the stratosphere on star image motion," Opt. Lett. 20, 1359-1361 (1995).
    [CrossRef] [PubMed]
  17. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, "Free space optical system performance for laser beam propagation through Non-Kolmogorov turbulence," Proc. SPIE 6457, 64570T-1-11 (2007).
  18. 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-1-12 (2007).
  19. 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, 661-672 (1993).
    [CrossRef]
  20. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, (SPIE Optical Engineering Press, Bellingham, 1998).
  21. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, (Dover Publications INC., New York, 1965).

2008 (1)

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 turbulence," Atmospheric Research 88, 66-77 (2008).
[CrossRef]

2004 (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, 175-177 (2000).
[CrossRef]

1999 (1)

M. S. Belen’kii, S. J. Karis, J. M. Brown, and R. Q. Fugate, "Experimental evidence of the effects of non-Kolmogorov turbulence and anisotropy of turbulence," Proc. SPIE 3749, 50-51 (1999).
[CrossRef]

1997 (1)

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]

1996 (3)

S. S. Moiseev and O. G. Chkhetiani, "Helical scaling in turbulence," JETP 83, 192-198 (1996).

T. Elperin, N. Kleeorin, and I. Rogachevskii, "Isotropic and anisotropic spectra of passive scalar fluctuations in turbulent fluid flow," Phys. Rev. E 53, 3431-3441 (1996).
[CrossRef]

G. D. Boreman and C. Dainty, "Zernike expansions for non-Kolmogorov turbulence," J. Opt. Soc. Am. A 13, 517-522 (1996).
[CrossRef]

1995 (4)

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 stratosphere on star image motion," Opt. Lett. 20, 1359-1361 (1995).
[CrossRef] [PubMed]

R. R. Beland, "Some aspects of propagation through weak isotropic non-Kolmogorov turbulence," Proc. SPIE 2375, 1111-1126 (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 (1)

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 2110, 43-55 (1994).
[CrossRef]

1993 (1)

1959 (1)

G. K. Batchelor, "Small-scale variation of convected quantities like temperature in turbulent fluid. Part I. General discussion and the case of small conductivity," J. Fluid Mech. 5, 113-133 (1959).
[CrossRef]

Andrews, L. C.

Batchelor, G. K.

G. K. Batchelor, "Small-scale variation of convected quantities like temperature in turbulent fluid. Part I. General discussion and the case of small conductivity," J. Fluid Mech. 5, 113-133 (1959).
[CrossRef]

Beland, R. R.

R. R. Beland, "Some aspects of propagation through weak isotropic non-Kolmogorov turbulence," Proc. SPIE 2375, 1111-1126 (1995).

Belen’kii, M. S.

M. S. Belen’kii, S. J. Karis, J. M. Brown, and R. Q. Fugate, "Experimental evidence of the effects of non-Kolmogorov turbulence and anisotropy of turbulence," Proc. SPIE 3749, 50-51 (1999).
[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, "Effect of the stratosphere on star image motion," Opt. Lett. 20, 1359-1361 (1995).
[CrossRef] [PubMed]

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]

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 2110, 43-55 (1994).
[CrossRef]

Boreman, G. D.

Brown, J. M.

M. S. Belen’kii, S. J. Karis, J. M. Brown, and R. Q. Fugate, "Experimental evidence of the effects of non-Kolmogorov turbulence and anisotropy of turbulence," Proc. SPIE 3749, 50-51 (1999).
[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]

Chkhetiani, O. G.

S. S. Moiseev and O. G. Chkhetiani, "Helical scaling in turbulence," JETP 83, 192-198 (1996).

Dainty, C.

Elperin, T.

T. Elperin, N. Kleeorin, and I. Rogachevskii, "Isotropic and anisotropic spectra of passive scalar fluctuations in turbulent fluid flow," Phys. Rev. E 53, 3431-3441 (1996).
[CrossRef]

Fugate, R. Q.

M. S. Belen’kii, S. J. Karis, J. M. Brown, and R. Q. Fugate, "Experimental evidence of the effects of non-Kolmogorov turbulence and anisotropy of turbulence," Proc. SPIE 3749, 50-51 (1999).
[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]

Golbraikh, E.

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 turbulence," Atmospheric Research 88, 66-77 (2008).
[CrossRef]

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

Gurvich, A. S.

Jiang, W.

C. Rao, W. Jiang, and N. Ling, "Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence," J. Mod. Opt. 47, 175-177 (2000).
[CrossRef]

Karis, S. J.

M. S. Belen’kii, S. J. Karis, J. M. Brown, and R. Q. Fugate, "Experimental evidence of the effects of non-Kolmogorov turbulence and anisotropy of turbulence," Proc. SPIE 3749, 50-51 (1999).
[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]

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 2110, 43-55 (1994).
[CrossRef]

Kleeorin, N.

T. Elperin, N. Kleeorin, and I. Rogachevskii, "Isotropic and anisotropic spectra of passive scalar fluctuations in turbulent fluid flow," Phys. Rev. E 53, 3431-3441 (1996).
[CrossRef]

Kopeika, N. S.

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 turbulence," Atmospheric Research 88, 66-77 (2008).
[CrossRef]

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

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 turbulence," Atmospheric Research 88, 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 2110, 43-55 (1994).
[CrossRef]

Ling, N.

C. Rao, W. Jiang, and N. Ling, "Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence," J. Mod. Opt. 47, 175-177 (2000).
[CrossRef]

Miller, W. B.

Moiseev, S. S.

S. S. Moiseev and O. G. Chkhetiani, "Helical scaling in turbulence," JETP 83, 192-198 (1996).

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 2110, 43-55 (1994).
[CrossRef]

Rao, C.

C. Rao, W. Jiang, and N. Ling, "Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence," J. Mod. Opt. 47, 175-177 (2000).
[CrossRef]

Ricklin, J. C.

Rogachevskii, I.

T. Elperin, N. Kleeorin, and I. Rogachevskii, "Isotropic and anisotropic spectra of passive scalar fluctuations in turbulent fluid flow," Phys. Rev. E 53, 3431-3441 (1996).
[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]

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 turbulence," Atmospheric Research 88, 66-77 (2008).
[CrossRef]

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]

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 turbulence," Atmospheric Research 88, 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 2110, 43-55 (1994).
[CrossRef]

Zilberman, 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 turbulence," Atmospheric Research 88, 66-77 (2008).
[CrossRef]

Appl. Opt. (1)

Atmospheric Research (1)

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 turbulence," Atmospheric Research 88, 66-77 (2008).
[CrossRef]

J. Fluid Mech. (1)

G. K. Batchelor, "Small-scale variation of convected quantities like temperature in turbulent fluid. Part I. General discussion and the case of small conductivity," J. Fluid Mech. 5, 113-133 (1959).
[CrossRef]

J. Mod. Opt. (1)

C. Rao, W. Jiang, and N. Ling, "Spatial and temporal characterization of phase fluctuations in non-Kolmogorov atmospheric turbulence," J. Mod. Opt. 47, 175-177 (2000).
[CrossRef]

J. Opt. Soc. Am. A (3)

JETP (1)

S. S. Moiseev and O. G. Chkhetiani, "Helical scaling in turbulence," JETP 83, 192-198 (1996).

Opt. Lett. (1)

Phys. Rev. E (1)

T. Elperin, N. Kleeorin, and I. Rogachevskii, "Isotropic and anisotropic spectra of passive scalar fluctuations in turbulent fluid flow," Phys. Rev. E 53, 3431-3441 (1996).
[CrossRef]

Proc. SPIE (5)

R. R. Beland, "Some aspects of propagation through weak isotropic non-Kolmogorov turbulence," Proc. SPIE 2375, 1111-1126 (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]

M. S. Belen’kii, S. J. Karis, J. M. Brown, and R. Q. Fugate, "Experimental evidence of the effects of non-Kolmogorov turbulence and anisotropy of turbulence," Proc. SPIE 3749, 50-51 (1999).
[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]

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 2110, 43-55 (1994).
[CrossRef]

Other (6)

G. Wang, "A new random-phase-screen time series simulation algorithm for dynamically atmospheric turbulence wave-front generator," Proc. SPIE 6027, 602716-1-12 (2006).

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, "Free space optical system performance for laser beam propagation through Non-Kolmogorov turbulence," Proc. SPIE 6457, 64570T-1-11 (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-1-12 (2007).

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, (SPIE Optical Engineering Press, Bellingham, 1998).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, (Dover Publications INC., New York, 1965).

A. Zilberman, E. Golbraikh, and N. S. Kopeika, "Lidar studies of aerosols and non-Kolmogorov turbulence in the Mediterranean troposphere," Proc. SPIE 5987, 598702-1-12 (2005).

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Figures (7)

Fig. 1.
Fig. 1.

The log-amplitude variance for a collimated beam (Θ0 = 1) as a function of power law α and the Fresnel ratio at the transmitter Λ0 for several values of the ratio ρ/W, with (a) for ρ/W = 0, (b) for ρ/W = 0.5, and (c) for ρ/W = 1.0.

Fig. 2.
Fig. 2.

The log-amplitude variance for a collimated beam (Θ0 = 1) as a function of power law α and the ratio ρ/W for several values of the Fresnel ratio at the transmitter Λ0, with (a) for Λ0 = 0.05, (b) for Λ0 = 1.00, and (c) for Λ0 = 100.

Fig. 3.
Fig. 3.

The log-amplitude variance for a divergent beam (Θ0 = 2) as a function of power law α and the Fresnel ratio at the transmitter Λ0 for several values of the ratio ρ/W, with (a) for ρ/W = 0, (b) for ρ/W = 0.5, and (c) for ρ/W = 1.0.

Fig. 4.
Fig. 4.

The log-amplitude variance for a divergent beam (Θ0 = 2) as a function of power law α and the ratio ρ/W for several values of the Fresnel ratio at the transmitter Λ0, with (a) for Λ0 = 0.05, (b) for Λ0 = 1.00, and (c) for Λ0 = 100.

Fig. 5.
Fig. 5.

The log-amplitude variance for a perfectly focus beam (Θ0 = 0) as a function of power law α and the Fresnel ratio at the transmitter Λ0 for several values of the ratio ρ/W, with (a) for ρ/W = 0, (b) for ρ/W = 0.5, and (c) for ρ/W = 1.0.

Fig. 6.
Fig. 6.

The on-axis log-amplitude variance as a function of power law α and the Fresnel ratio at the transmitter Λ0 for several curvature parameters at the transmitter Θ0, with (a) for Θ0 = 0.01, (b) for Θ0 = 0.10, and (c) for Θ0 = 0.50.

Fig. 7.
Fig. 7.

The diffractive edge (ρ/W = 1) log-amplitude variance as a function of power law α and the Fresnel ratio at the transmitter Λ0 for several curvature parameters at the transmitter Θ0, with (a) for Θ0 = 0.01, (b) for Θ0 = 0.10, and (c) for Θ0 = 0.50.

Tables (1)

Tables Icon

Table 1. Expressions of Log-amplitude Variance for Various Beam Types

Equations (22)

Equations on this page are rendered with MathJax. Learn more.

Φ n ( κ , α ) = A ( α ) C n 2 ̃ κ α , 2 π / L 0 κ 2 π / l 0 , 3 < α < 4 ,
A ( α ) = 1 4 π 2 Γ ( α 1 ) cos ( α π 2 ) ,
Φ n ( κ ) = 0.033 C n 2 κ 11 / 3 ,
σ χ 2 ( ρ ) = 2 π 2 k 2 L 0 1 0 κ Φ n ( κ ) { I 0 ( 2 Λ ρξκ ) cos [ L κ 2 k ( 1 Θ ̃ ξ ) ξ ] } × [ exp ( Λ L ξ 2 κ 2 k ) ] dκdξ ,
Θ = Θ 0 Θ 0 2 + Λ 0 2 , Λ = Λ 0 Θ 0 2 + Λ 0 2 ,
σ χ 2 ( ρ ) = σ χ , r 2 ( ρ ) + σ χ , l 2 ,
σ χ , r 2 ( ρ ) = 2 π 2 k 2 L 0 1 0 κ Φ n ( κ ) [ I 0 ( 2 Λ ρξκ ) 1 ] × [ exp ( Λ L ξ 2 κ 2 k ) ] dκdξ ,
σ χ , l 2 = 4 π 2 k 2 L 0 1 0 κ Φ n ( κ ) sin 2 [ L κ 2 2 k ( 1 Θ ̃ ξ ) ξ ] × [ exp ( Λ L ξ 2 κ 2 k ) ] dκdξ .
σ χ , r 2 ( ρ , α ) = 1 α 1 A ( α ) C ̃ n 2 π 2 k 3 α / 2 L α / 2 Γ ( 2 α 2 ) Λ α 2 2 × [ 1 F 1 ( 2 α 2 ; 1 ; 2 ρ 2 W 2 ) 1 ] ,
σ χ , r 2 ( ρ , α ) = 2 α 1 A ( α ) C ̃ n 2 π 2 k 3 α / 2 L α / 2 Λ α 2 2 ρ 2 W 2 × [ Γ ( 4 α 2 ) + Γ ( 6 α 2 ) ρ 2 2 W 2 ] , ρ W ,
σ χ , r 2 ( ρ , α ) = 1 α 1 2 α / 2 A ( α ) C ̃ n 2 π 2 k 3 a / 2 L a / 2 Λ α 2 2 × ( W ρ ) α exp ( 2 ρ 2 W 2 ) , ρ W .
σ χ , l 2 ( α ) = 1 α 1 A ( α ) C ̃ n 2 π 2 k 3 a / 2 L a / 2 Γ ( 2 α 2 ) { Λ α 2 2 Re [ 2 α 2 α i α 2 2 2 F 1 ( 2 α 2 , α 2 ; 2 + α 2 ; ( Θ ̃ + i Λ ) ) ] } ,
σ χ 2 ( ρ , α ) = 1 α 1 A ( α ) C ̃ n 2 π 2 k 3 a / 2 L a / 2 Γ ( 2 α 2 ) { Λ α 2 2 1 F 1 ( 2 α 2 ; 1 ; 2 ρ 2 W 2 ) Re [ 2 α 2 α i α 2 2 2 F 1 ( 2 α 2 , α 2 ; 2 + α 2 ; ( Θ ̃ + i Λ ) ) ] } .
Re [ 2 α 2 α i α 2 2 2 F 1 ( 2 α 2 , α 2 ; 2 + α 2 ; ( Θ ̃ + i Λ ) ) ] = 2 α 2 2 n = 0 ( 2 α 2 ) n ( α 2 ) n ( 2 + α 2 ) n n ! ( Θ 2 ̃ + Λ 2 ) n / 2 cos [ n tan 1 ( Λ Θ ̃ ) + ( α 2 ) π 4 ] ,
2 F 1 ( a , b ; c ; y ) = Γ ( c ) Γ ( b a ) Γ ( b ) Γ ( c a ) ( y ) a 2 F 1 ( a , 1 c + a ; 1 b + a ; 1 y ) + Γ ( c ) Γ ( a b ) Γ ( a ) Γ ( c b ) ( y ) b 2 F 1 ( b , 1 c + b ; 1 a + b ; 1 y ) ,
Re [ 2 α 2 α i α 2 2 2 F 1 ( 2 α 2 , α 2 ; 2 + α 2 ; ( Θ ̃ + i Λ ) ) ] = ( Θ 2 ̃ + Λ 2 ) α 2 4 n = 0 ( 2 α 2 ) n ( 1 α ) n ( 2 α ) n n! ( Θ 2 ̃ + Λ 2 ) n / 2 cos [ ( n α 2 2 ) × tan 1 ( Λ Θ ̃ ) + ( α 2 ) π 4 ] + 2 α 2 α Γ ( 1 + α / 2 ) Γ ( 1 α ) Γ ( 1 α / 2 ) ( Θ 2 ̃ + Λ 2 ) α 4 × cos [ α 2 tan 1 ( Λ Θ ̃ ) + ( 2 3 α ) π 4 ] .
σ χ , l 2 ( α ) = 1 α 1 A ( α ) C ̃ n 2 π 2 k 3 a / 2 L a / 2 Γ ( 2 α 2 ) { ( Λ 0 1 + Λ 0 2 ) α 2 2 2 α 2 α × n = 0 ( 2 α 2 ) n ( α 2 ) n ( 2 + α 2 ) n n ! ( Λ 0 2 1 + Λ 0 2 ) ) n / 2 cos [ n tan 1 ( 1 Λ 0 ) + ( α 2 ) π 4 ] } .
σ χ , p 2 ( α ) = 2 α A ( α ) C ̃ n 2 π 2 k 3 a / 2 L a / 2 Γ ( 2 α 2 ) cos [ ( α 2 ) π 4 ] ,
σ χ , s 2 ( α ) = 2 α A ( α ) C ̃ n 2 π 2 k 3 a / 2 L a / 2 Γ ( α 2 ) Γ ( 2 α 2 ) Γ ( 2 + α 2 ) Γ ( α ) cos [ ( α 2 ) π 4 ] .
σ χ , l 2 ( ρ , α ) = 1 α 1 A ( α ) C ̃ n 2 π 2 k 3 a / 2 L a / 2 Γ ( 2 α 2 ) × { Λ α 2 2 2 α 2 α n = 0 ( 2 α 2 ) n ( α 2 ) n ( 2 + α 2 ) n n ! ( Θ 2 ̃ + Λ 2 ) n / 2 cos [ n tan 1 ( Λ Θ ̃ ) + ( α 2 ) π 4 ] } .
σ χ , l 2 ( ρ , α ) = 1 α 1 A ( α ) C ̃ n 2 π 2 k 3 a / 2 L a / 2 Γ ( 2 α 2 ) { Λ α 2 2 ( Θ 2 ̃ + Λ 2 ) α 2 4 × n = 0 ( 2 α 2 ) n ( 1 α ) n ( 2 α ) n n ! ( Θ 2 ̃ + Λ 2 ) n / 2 cos [ ( n α 2 2 ) tan 1 ( Λ Θ ̃ ) + ( α 2 ) π 4 ] 2 α α α Γ ( 1 + α / 2 ) Γ ( 1 α ) Γ ( 1 α / 2 ) ( Θ 2 ̃ + Λ 2 ) α 4 × cos [ α 2 tan 1 ( Λ Θ ̃ ) + ( 2 3 α ) π 4 ] .
σ χ , l 2 ( ρ , α ) = 1 α 1 A ( α ) C ̃ n 2 π 2 k 3 a / 2 L a / 2 Γ ( 2 α 2 ) { Λ 0 2 α 2 ( 1 + Λ 0 2 Λ 0 2 ) α 2 4 × n = 0 ( 2 α 2 ) n ( 1 α ) n ( 2 α ) n n ! ( Λ 0 2 1 + Λ 0 2 ) n / 2 cos [ ( n α 2 2 ) tan 1 ( 1 Λ 0 ) + ( α 2 ) π 4 ] 2 α 2 α Γ ( 1 + α / 2 ) Γ ( 1 α ) Γ ( 1 α / 2 ) ( Λ 0 2 1 + Λ 0 2 ) α 4 × cos [ α 2 tan 1 ( 1 Λ 0 ) + ( 2 3 α ) π 4 ] } .

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