Abstract

We consider the point-spread function (PSF) of an imaging system looking through weak optical turbulence along a horizontal path through the atmospheric surface layer. From this PSF we derive a scalar total irradiance field and a center-of-mass vector field. Theoretical values are found for the space–time autocorrelation and cross-correlation functions for these fields, which are then compared with the observed correlation functions obtained from data taken at the Validation Measurements on Propagation in the Infrared and Radar (VAMPIRA) measurement trial. We discuss the meaning of these results and possible directions for future work.

© 2007 Optical Society of America

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References

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  1. V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).
  2. J. H. Shapiro, "Imaging and optical communication through atmospheric turbulence," in Laser Beam Propagation in the Atmosphere, J.W.Strohbehn, ed. (Springer-Verlag, 1978), pp. 171-222.
  3. R. W. Lee and J. C. Harp, "Weak scattering in random media, with applications to remote probing," Proc. IEEE 57, 375-406 (1969).
    [CrossRef]
  4. R. J. Noll, "Zernike polynomials and atmospheric turbulence," J. Opt. Soc. Am. 66, 207-211 (1976).
    [CrossRef]
  5. N. Roddier, "Atmospheric wavefront simulation using Zernike polynomials," Opt. Eng. 29, 1174-1180 (1990).
    [CrossRef]
  6. D. L. Fried, "Optical resolution through a randomly inhomogeneous medium for very long and very short exposures," J. Opt. Soc. Am. 56, 1372-1379 (1966).
    [CrossRef]
  7. R. R. Beland, "Propagation through atmospheric optical turbulence," in Atmospheric Propagation of Radiation, F.G.Smith, ed. Vol. 2 of The Infrared and Electro-Optical Systems Handbook (SPIE, 1993), pp. 157-232.
  8. M. S. Belenkii, J. M. Steward, and P. Gillespie, "Turbulence-induced edge image waviness: theory and experiment," Appl. Opt. 40, 1321-1328 (2001).
    [CrossRef]
  9. M. S. Belenkii, K. Burgi, P. Gillespie, G. G. Gimmestad, K. Hughes, D. Rogerts, J. Stewart, and J. Wood, "Measurement of tilt anisoplanatism with discrete and extended sources," in Free-Space Laser Communication and Active Laser Illumination III, D. G. Voelz and J. C. Ricklin, eds., Proc. SPIE 5160, 107-120 (2004).
    [CrossRef]
  10. D. M. Winker, "Effect of a finite outer scale on the Zernike decomposition of atmospheric optical turbulence," J. Opt. Soc. Am. A 8, 1568-1573 (1991).
    [CrossRef]
  11. G. D. Boreman and C. Dainty, "Zernike expansions for non-kolmogorov turbulence," J. Opt. Soc. Am. A 13, 517-522 (1996).
    [CrossRef]
  12. M. S. Belenkii, 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," in Adaptive Optics Systems and Technology, R. K. Tyson and R. Q. Fugate, eds., Proc. SPIE 3762, 396-406 (1999).
    [CrossRef]
  13. H. J. M. Heemskerk, "VAMPIRA--radar and infrared propagation synergism trial," in Proceedings of European Conference on Propagation and Systems, Brest, France, March 15-18, 2005.
  14. S. F. Clifford, "Temporal-frequency spectra for a spherical wave propagating through atmospheric turbulence," J. Opt. Soc. Am. 61, 1285-1292 (1971).
    [CrossRef]
  15. P.-Ch. Grosse, "Analyse des effets de la turbulence sur l'imagerie optronique," Internship Rep. (Écoles Militaires de Saint-Cyr Coëtquidan, France, 2004).
  16. J. C. Wyngaard and S. F. Clifford, "Taylor's hypothesis and high-frequency turbulence spectra," J. Atmos. Sci. 34, 922-929 (1977).
    [CrossRef]
  17. R. Conan, J. Borgnino, A. Ziad, and F. Martin, "Analytical solution for the covariance and for the decorrelation time of the angle of arrival of a wave front corrugated by atmospheric turbulence," J. Opt. Soc. Am. A 17, 1807-1818 (2000).
    [CrossRef]
  18. R. C. Warren, "A phenomenological model of scintillation of infrared radiation from point targets over water and measurements of the model parameters," DSTO-RR-0231 (Defence Science and Technology Organisation Aeronautical and Maritime Research Laboratory, Australia, 2002).
  19. D. H. Tofsted and S. G. O'Brien, "Simulation of atmospheric turbulence image distortion and scintillation effects impacting short-wave infrared (SWIR) active imaging systems," in Targets and Backgrounds X: Characterization and Representation, W. R. Watkins, D. Clement and W. R. Reynolds, eds., Proc. SPIE 5431, 160-171 (2004).
  20. J. Recolons and F. Dios, "Accurate calculation of phase screens for the modelling of leaser beam propagation through atmospheric turbulence," in Atmospheric Optical Modeling, Measurement, and Simulation, S. M. Doss-Hammel and A. Kohnle, eds., Proc. SPIE 5891, 589107 (2005).

2004

M. S. Belenkii, K. Burgi, P. Gillespie, G. G. Gimmestad, K. Hughes, D. Rogerts, J. Stewart, and J. Wood, "Measurement of tilt anisoplanatism with discrete and extended sources," in Free-Space Laser Communication and Active Laser Illumination III, D. G. Voelz and J. C. Ricklin, eds., Proc. SPIE 5160, 107-120 (2004).
[CrossRef]

2001

2000

1999

M. S. Belenkii, 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," in Adaptive Optics Systems and Technology, R. K. Tyson and R. Q. Fugate, eds., Proc. SPIE 3762, 396-406 (1999).
[CrossRef]

1996

1991

1990

N. Roddier, "Atmospheric wavefront simulation using Zernike polynomials," Opt. Eng. 29, 1174-1180 (1990).
[CrossRef]

1977

J. C. Wyngaard and S. F. Clifford, "Taylor's hypothesis and high-frequency turbulence spectra," J. Atmos. Sci. 34, 922-929 (1977).
[CrossRef]

1976

1971

1969

R. W. Lee and J. C. Harp, "Weak scattering in random media, with applications to remote probing," Proc. IEEE 57, 375-406 (1969).
[CrossRef]

1966

Barchers, J. D.

M. S. Belenkii, 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," in Adaptive Optics Systems and Technology, R. K. Tyson and R. Q. Fugate, eds., Proc. SPIE 3762, 396-406 (1999).
[CrossRef]

Beland, R. R.

R. R. Beland, "Propagation through atmospheric optical turbulence," in Atmospheric Propagation of Radiation, F.G.Smith, ed. Vol. 2 of The Infrared and Electro-Optical Systems Handbook (SPIE, 1993), pp. 157-232.

Belenkii, M. S.

M. S. Belenkii, K. Burgi, P. Gillespie, G. G. Gimmestad, K. Hughes, D. Rogerts, J. Stewart, and J. Wood, "Measurement of tilt anisoplanatism with discrete and extended sources," in Free-Space Laser Communication and Active Laser Illumination III, D. G. Voelz and J. C. Ricklin, eds., Proc. SPIE 5160, 107-120 (2004).
[CrossRef]

M. S. Belenkii, J. M. Steward, and P. Gillespie, "Turbulence-induced edge image waviness: theory and experiment," Appl. Opt. 40, 1321-1328 (2001).
[CrossRef]

M. S. Belenkii, 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," in Adaptive Optics Systems and Technology, R. K. Tyson and R. Q. Fugate, eds., Proc. SPIE 3762, 396-406 (1999).
[CrossRef]

Boreman, G. D.

Borgnino, J.

Brown, J. M.

M. S. Belenkii, 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," in Adaptive Optics Systems and Technology, R. K. Tyson and R. Q. Fugate, eds., Proc. SPIE 3762, 396-406 (1999).
[CrossRef]

Burgi, K.

M. S. Belenkii, K. Burgi, P. Gillespie, G. G. Gimmestad, K. Hughes, D. Rogerts, J. Stewart, and J. Wood, "Measurement of tilt anisoplanatism with discrete and extended sources," in Free-Space Laser Communication and Active Laser Illumination III, D. G. Voelz and J. C. Ricklin, eds., Proc. SPIE 5160, 107-120 (2004).
[CrossRef]

Clifford, S. F.

J. C. Wyngaard and S. F. Clifford, "Taylor's hypothesis and high-frequency turbulence spectra," J. Atmos. Sci. 34, 922-929 (1977).
[CrossRef]

S. F. Clifford, "Temporal-frequency spectra for a spherical wave propagating through atmospheric turbulence," J. Opt. Soc. Am. 61, 1285-1292 (1971).
[CrossRef]

Conan, R.

Dainty, C.

Dios, F.

J. Recolons and F. Dios, "Accurate calculation of phase screens for the modelling of leaser beam propagation through atmospheric turbulence," in Atmospheric Optical Modeling, Measurement, and Simulation, S. M. Doss-Hammel and A. Kohnle, eds., Proc. SPIE 5891, 589107 (2005).

Fried, D. L.

Fugate, R. Q.

M. S. Belenkii, 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," in Adaptive Optics Systems and Technology, R. K. Tyson and R. Q. Fugate, eds., Proc. SPIE 3762, 396-406 (1999).
[CrossRef]

Gillespie, P.

M. S. Belenkii, K. Burgi, P. Gillespie, G. G. Gimmestad, K. Hughes, D. Rogerts, J. Stewart, and J. Wood, "Measurement of tilt anisoplanatism with discrete and extended sources," in Free-Space Laser Communication and Active Laser Illumination III, D. G. Voelz and J. C. Ricklin, eds., Proc. SPIE 5160, 107-120 (2004).
[CrossRef]

M. S. Belenkii, J. M. Steward, and P. Gillespie, "Turbulence-induced edge image waviness: theory and experiment," Appl. Opt. 40, 1321-1328 (2001).
[CrossRef]

Gimmestad, G. G.

M. S. Belenkii, K. Burgi, P. Gillespie, G. G. Gimmestad, K. Hughes, D. Rogerts, J. Stewart, and J. Wood, "Measurement of tilt anisoplanatism with discrete and extended sources," in Free-Space Laser Communication and Active Laser Illumination III, D. G. Voelz and J. C. Ricklin, eds., Proc. SPIE 5160, 107-120 (2004).
[CrossRef]

Grosse, P.-Ch.

P.-Ch. Grosse, "Analyse des effets de la turbulence sur l'imagerie optronique," Internship Rep. (Écoles Militaires de Saint-Cyr Coëtquidan, France, 2004).

Harp, J. C.

R. W. Lee and J. C. Harp, "Weak scattering in random media, with applications to remote probing," Proc. IEEE 57, 375-406 (1969).
[CrossRef]

Heemskerk, H. J. M.

H. J. M. Heemskerk, "VAMPIRA--radar and infrared propagation synergism trial," in Proceedings of European Conference on Propagation and Systems, Brest, France, March 15-18, 2005.

Hughes, K.

M. S. Belenkii, K. Burgi, P. Gillespie, G. G. Gimmestad, K. Hughes, D. Rogerts, J. Stewart, and J. Wood, "Measurement of tilt anisoplanatism with discrete and extended sources," in Free-Space Laser Communication and Active Laser Illumination III, D. G. Voelz and J. C. Ricklin, eds., Proc. SPIE 5160, 107-120 (2004).
[CrossRef]

Karis, S. J.

M. S. Belenkii, 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," in Adaptive Optics Systems and Technology, R. K. Tyson and R. Q. Fugate, eds., Proc. SPIE 3762, 396-406 (1999).
[CrossRef]

Lee, R. W.

R. W. Lee and J. C. Harp, "Weak scattering in random media, with applications to remote probing," Proc. IEEE 57, 375-406 (1969).
[CrossRef]

Martin, F.

Noll, R. J.

O'Brien, S. G.

D. H. Tofsted and S. G. O'Brien, "Simulation of atmospheric turbulence image distortion and scintillation effects impacting short-wave infrared (SWIR) active imaging systems," in Targets and Backgrounds X: Characterization and Representation, W. R. Watkins, D. Clement and W. R. Reynolds, eds., Proc. SPIE 5431, 160-171 (2004).

Osmon, C. L.

M. S. Belenkii, 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," in Adaptive Optics Systems and Technology, R. K. Tyson and R. Q. Fugate, eds., Proc. SPIE 3762, 396-406 (1999).
[CrossRef]

Recolons, J.

J. Recolons and F. Dios, "Accurate calculation of phase screens for the modelling of leaser beam propagation through atmospheric turbulence," in Atmospheric Optical Modeling, Measurement, and Simulation, S. M. Doss-Hammel and A. Kohnle, eds., Proc. SPIE 5891, 589107 (2005).

Roddier, N.

N. Roddier, "Atmospheric wavefront simulation using Zernike polynomials," Opt. Eng. 29, 1174-1180 (1990).
[CrossRef]

Rogerts, D.

M. S. Belenkii, K. Burgi, P. Gillespie, G. G. Gimmestad, K. Hughes, D. Rogerts, J. Stewart, and J. Wood, "Measurement of tilt anisoplanatism with discrete and extended sources," in Free-Space Laser Communication and Active Laser Illumination III, D. G. Voelz and J. C. Ricklin, eds., Proc. SPIE 5160, 107-120 (2004).
[CrossRef]

Shapiro, J. H.

J. H. Shapiro, "Imaging and optical communication through atmospheric turbulence," in Laser Beam Propagation in the Atmosphere, J.W.Strohbehn, ed. (Springer-Verlag, 1978), pp. 171-222.

Steward, J. M.

Stewart, J.

M. S. Belenkii, K. Burgi, P. Gillespie, G. G. Gimmestad, K. Hughes, D. Rogerts, J. Stewart, and J. Wood, "Measurement of tilt anisoplanatism with discrete and extended sources," in Free-Space Laser Communication and Active Laser Illumination III, D. G. Voelz and J. C. Ricklin, eds., Proc. SPIE 5160, 107-120 (2004).
[CrossRef]

Tatarski, V. I.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).

Tofsted, D. H.

D. H. Tofsted and S. G. O'Brien, "Simulation of atmospheric turbulence image distortion and scintillation effects impacting short-wave infrared (SWIR) active imaging systems," in Targets and Backgrounds X: Characterization and Representation, W. R. Watkins, D. Clement and W. R. Reynolds, eds., Proc. SPIE 5431, 160-171 (2004).

Warren, R. C.

R. C. Warren, "A phenomenological model of scintillation of infrared radiation from point targets over water and measurements of the model parameters," DSTO-RR-0231 (Defence Science and Technology Organisation Aeronautical and Maritime Research Laboratory, Australia, 2002).

Winker, D. M.

Wood, J.

M. S. Belenkii, K. Burgi, P. Gillespie, G. G. Gimmestad, K. Hughes, D. Rogerts, J. Stewart, and J. Wood, "Measurement of tilt anisoplanatism with discrete and extended sources," in Free-Space Laser Communication and Active Laser Illumination III, D. G. Voelz and J. C. Ricklin, eds., Proc. SPIE 5160, 107-120 (2004).
[CrossRef]

Wyngaard, J. C.

J. C. Wyngaard and S. F. Clifford, "Taylor's hypothesis and high-frequency turbulence spectra," J. Atmos. Sci. 34, 922-929 (1977).
[CrossRef]

Ziad, A.

Appl. Opt.

J. Atmos. Sci.

J. C. Wyngaard and S. F. Clifford, "Taylor's hypothesis and high-frequency turbulence spectra," J. Atmos. Sci. 34, 922-929 (1977).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Eng.

N. Roddier, "Atmospheric wavefront simulation using Zernike polynomials," Opt. Eng. 29, 1174-1180 (1990).
[CrossRef]

Proc. IEEE

R. W. Lee and J. C. Harp, "Weak scattering in random media, with applications to remote probing," Proc. IEEE 57, 375-406 (1969).
[CrossRef]

Proc. SPIE

M. S. Belenkii, K. Burgi, P. Gillespie, G. G. Gimmestad, K. Hughes, D. Rogerts, J. Stewart, and J. Wood, "Measurement of tilt anisoplanatism with discrete and extended sources," in Free-Space Laser Communication and Active Laser Illumination III, D. G. Voelz and J. C. Ricklin, eds., Proc. SPIE 5160, 107-120 (2004).
[CrossRef]

M. S. Belenkii, 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," in Adaptive Optics Systems and Technology, R. K. Tyson and R. Q. Fugate, eds., Proc. SPIE 3762, 396-406 (1999).
[CrossRef]

Other

H. J. M. Heemskerk, "VAMPIRA--radar and infrared propagation synergism trial," in Proceedings of European Conference on Propagation and Systems, Brest, France, March 15-18, 2005.

P.-Ch. Grosse, "Analyse des effets de la turbulence sur l'imagerie optronique," Internship Rep. (Écoles Militaires de Saint-Cyr Coëtquidan, France, 2004).

R. R. Beland, "Propagation through atmospheric optical turbulence," in Atmospheric Propagation of Radiation, F.G.Smith, ed. Vol. 2 of The Infrared and Electro-Optical Systems Handbook (SPIE, 1993), pp. 157-232.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).

J. H. Shapiro, "Imaging and optical communication through atmospheric turbulence," in Laser Beam Propagation in the Atmosphere, J.W.Strohbehn, ed. (Springer-Verlag, 1978), pp. 171-222.

R. C. Warren, "A phenomenological model of scintillation of infrared radiation from point targets over water and measurements of the model parameters," DSTO-RR-0231 (Defence Science and Technology Organisation Aeronautical and Maritime Research Laboratory, Australia, 2002).

D. H. Tofsted and S. G. O'Brien, "Simulation of atmospheric turbulence image distortion and scintillation effects impacting short-wave infrared (SWIR) active imaging systems," in Targets and Backgrounds X: Characterization and Representation, W. R. Watkins, D. Clement and W. R. Reynolds, eds., Proc. SPIE 5431, 160-171 (2004).

J. Recolons and F. Dios, "Accurate calculation of phase screens for the modelling of leaser beam propagation through atmospheric turbulence," in Atmospheric Optical Modeling, Measurement, and Simulation, S. M. Doss-Hammel and A. Kohnle, eds., Proc. SPIE 5891, 589107 (2005).

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Figures (11)

Fig. 1
Fig. 1

Simplified representation of an imaging system in atmospheric turbulence.

Fig. 2
Fig. 2

Map of Eckernforde Bay featuring the two propagation paths (arrows) used during the VAMPIRA trial.

Fig. 3
Fig. 3

Schematic representation of the light setup at the Bookniseck site along with an inset photograph of the setup.

Fig. 4
Fig. 4

Sample image from the high-speed digital camera sequence taken at 16h01 GMT, March 29, 2004. The white markers indicate the actual positions of the lights.

Fig. 5
Fig. 5

Autocorrelation and cross-correlation functions for the log-irradiance at 04h00 GMT, April 1, 2004. Shown are the theoretical and observed autocorrelation functions as well as the cross-correlation functions for lights 0.5 m apart and 1 m apart. The normalized time interval is obtained using the effective wind speed: τ * = U e τ μ .

Fig. 6
Fig. 6

Same as in Fig. 5 but for the total center-of-mass displacements, with the outer scale set to 11 m .

Fig. 7
Fig. 7

PWF for the variances of the scintillation (black curve) and the total center of mass (gray curve) as a function of the normalized path coordinate ζ. The source is located at ζ = 0 , the receiver is at ζ = 1 , and the PWFs are normalized so that their maxima equal 1.

Fig. 8
Fig. 8

Simplified illustration of the physical cause for the difference between the log-irradiance and the center-of-mass correlation functions. The imaging system (to the right) observes two sources, labeled A and B, each with a line of sight represented by a dashed arrow. The small gray circle labeled S represents the turbulent eddy most responsible for the scintillation, and the large gray circle C is the turbulent eddy most responsible for the center-of-mass displacements. The solid black arrows represent the distance each eddy must travel to go from the line of sight of A to that of B.

Fig. 9
Fig. 9

Same as in Fig. 6 but for the horizontal ( X ) component of the center-of-mass displacement.

Fig. 10
Fig. 10

Same as in Fig. 6 but for the vertical ( Y ) component of the center-of-mass displacement.

Fig. 11
Fig. 11

Joint correlation functions between the log-irradiance and the horizontal ( X ) component of the center-of-mass displacement.

Equations (54)

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

ψ ( ρ l , ρ o ) = k 2 L 2 π n ( r ) ( L z ) z exp [ i k L 2 z ( L z ) ( ρ z L ρ l L z L ρ o ) 2 ] d 3 r ,
ψ ( ρ l , ρ o ) = k 2 2 π 0 1 d η d 2 ρ n ( ρ , L η ) η ( 1 η ) exp [ i π η ( 1 η ) ( ρ η ρ l ( 1 η ) ρ o μ ) 2 ] .
n ( r , t ) = n ( x U x t , y , z ) ,
n ( r , t ) = d 3 K exp { i [ K x ( x U x t ) + K y y + K z z ] } N ( K ) .
q = π η ( 1 η ) ( ρ η ρ l ( 1 η ) ρ o μ )
ψ ( ρ l , ρ o , t ) = k L π 0 1 d η d 3 K N ( K ) exp [ i ( K T γ + K z L η K x U x t ) ] F ( K T , η ) ,
F ( K T , η ) = d 2 q exp [ i q 2 + i μ K T q η ( 1 η ) π ] ,
F ( K T , η ) = i π exp [ i K T 2 μ 2 η ( 1 η ) 4 π ] .
χ ( ρ l , ρ o , t ) = k L 0 1 d η d 3 K N ( K ) exp [ i ( K T γ + K z L η K x U x t ) ] sin [ K T 2 μ 2 η ( 1 η ) 4 π ] ,
S ( ρ l , ρ o , t ) = k L 0 1 d η d 3 K N ( K ) exp [ i ( K T γ + K z L η K x U x t ) ] cos [ K T 2 μ 2 η ( 1 η ) 4 π ] .
I i ( α i , t ) = d 2 α o P ( α o , α i , t ) I o ( α o , t ) ,
P ( α o , α i , t ) = k 2 4 π 3 f 2 d 2 ρ l d 2 ρ l W ( ρ l ) W ( ρ l ) exp [ i k ( ρ l ρ l ) ( α i α o ) + ψ ( ρ l , L α o , t ) + ψ * ( ρ l , L α o , t ) ] ,
M 0 ( a o , t ) = I * ( a o , α i , t ) d 2 α i ,
M 1 ( a o , t ) = α i I * ( a o , α i , t ) d 2 α i
C ( a o , t ) = M 1 ( a o , t ) M 0 ( a o , t ) .
k 2 4 π 2 exp [ i k ( ρ l ρ l ) α i ] d 2 α i = δ ( ρ l ρ l )
M 0 ( a o , t ) = 1 π f 2 d 2 ρ l W ( ρ l ) exp [ 2 χ ( ρ l , L a o , t ) ] ,
i k 3 4 π 2 ( α i a o ) exp [ i k ( ρ l ρ l ) α i ] d 2 α i = δ ( ρ l ρ l )
C ( a o , t ) = a o d 2 ρ l W ( ρ l ) exp [ 2 χ ( ρ l , L a o , t ) ] l S ( ρ l , L a o , t ) k d 2 ρ l W ( ρ l ) exp [ 2 χ ( ρ l , L a o , t ) ] ,
M 0 ( a o , t ) D 2 4 f 2 [ 1 + 8 π D 2 d 2 ρ l W ( ρ l ) χ ( ρ l , L a o , t ) ] .
H ( a o , t ) ln ( D 2 4 f 2 ) + 8 π D 2 d 2 ρ l W ( ρ l ) χ ( ρ l , L a o , t ) ,
h ( a o , t ) 8 π D 2 0 D 2 ρ l d ρ l 0 2 π d ϕ χ ( ρ l , L a o , t ) .
J 0 ( x ) = 1 2 π 0 2 π exp ( i x cos ϕ ) d ϕ ,
h ( a o , t ) 16 k L D 2 0 D 2 ρ l d ρ l 0 1 d η d 3 K N ( K ) J 0 ( K T η ρ l ) × exp [ i K T a o L ( 1 η ) + i K z η L i K x U x t ] sin [ K T 2 μ 2 η ( 1 η ) 4 π ] .
0 x x J 0 ( a x ) d x = x a J 1 ( a x ) ,
h ( a o , t ) 2 k L 0 1 d η d 3 K N ( K ) β ( K T η D 2 ) × exp [ i K T a o L ( 1 η ) + i K z η L i K x U x t ] sin [ K T 2 μ 2 η ( 1 η ) 4 π ] ,
B h h ( a o , t , a o , t ) = h ( a o , t ) h ( a o , t ) ¯ ,
N ( K ) N * ( K ) ¯ = Φ ( K + K 2 ) δ ( K K ) ,
B h h ( a o , t , a o , t ) = 4 k 2 L 2 0 1 d η 0 1 d η d 3 K Φ ( K ) exp [ i K z L ( η η ) ] exp [ i K x U ( t t ) ] exp [ i L K T { ( 1 η ) a o ( 1 η ) a o } ] β ( K T η D 2 ) β ( K T η D 2 ) × sin [ K T 2 μ 2 η ( 1 η ) 4 π ] sin [ K T 2 μ 2 η ( 1 η ) 4 π ] .
B h h ( a o , a o , τ ) = 4 k 2 L 2 1 1 d ϵ ϵ 2 1 ϵ 2 d ζ d 3 K Φ ( K ) exp ( i K z L ϵ ) exp ( i K x U τ ) β ( K T { ζ + ϵ 2 } D 2 ) β ( K T { ζ ϵ 2 } D 2 ) exp [ i L K T { ( 1 ζ ) ( a o a o ) ( a o + a o ) ϵ 2 } ] sin [ K T 2 μ 2 ( ζ + ϵ 2 ) ( 1 ζ ϵ 2 ) 4 π ] sin [ K T 2 μ 2 ( ζ ϵ 2 ) ( 1 ζ + ϵ 2 ) 4 π ] .
L 2 π exp ( i K z L ϵ ) d ϵ = δ ( K z ) ,
B h h ( a o a o , τ ) 8 π k 2 L 0 1 d ζ d 2 K T Φ ( K T , 0 ) exp [ i L K T ( 1 ζ ) ( a o a o ) i K x U τ ] β 2 ( K T ζ D 2 ) sin 2 [ K T 2 μ 2 ζ ( 1 ζ ) 4 π ] .
B h h ( Δ o , τ ) 16 π 2 k 2 L 0 1 d ζ 0 d K K Φ ( K ) β 2 ( K ζ D 2 ) × J 0 ( K ( 1 ζ ) Δ o U τ ) sin 2 [ K 2 μ 2 ζ ( 1 ζ ) 4 π ] ,
c ( a o , t ) = d 2 ρ l W ( ρ l ) exp [ 2 χ ( ρ l , L a o , t ) ] l S ( ρ l , L a o , t ) k d 2 ρ l W ( ρ l ) exp [ 2 χ ( ρ l , L a o , t ) ] .
c ( a o , t ) 4 π k D 2 d 2 ρ l W ( ρ l ) l S ( ρ l , L a o , t ) .
c ( a o , t ) i L 0 1 d η d 3 K N ( K ) K T η β ( K T η D 2 ) exp [ i ( K T a o L ( 1 η ) + K z L η K x U x t ) ] cos [ K T 2 μ 2 η ( 1 η ) 4 π ] .
B c c ( a o , t , a o , t ) = c ( a o , t ) c ( a o , t ) ¯ .
B c c ( Δ o , τ ) 4 π 2 L 0 1 d ζ 0 d K K 3 ζ 2 Φ ( K ) β 2 ( K ζ D 2 ) J 0 ( K ( 1 ζ ) Δ o U τ ) cos 2 [ K 2 μ 2 ζ ( 1 ζ ) 4 π ] .
B x x ( a o , t , a o , t ) = x ( a o , t ) x ( a o , t ) ¯ .
B x x ( Δ o , τ ) 2 π L 0 1 d ζ 0 2 π d ϕ 0 d K K 3 ζ 2 Φ ( K ) β 2 ( K ζ D 2 ) exp [ i K ( 1 ζ ) Δ o U τ cos ( ϕ θ ) ] cos 2 ( ϕ ) cos 2 [ K 2 μ 2 ζ ( 1 ζ ) 4 π ] ,
I ( α , θ ) = 0 2 π d ϕ cos 2 ( ϕ ) exp [ i α cos ( ϕ θ ) ] .
I ( α , θ ) = π J 0 ( α ) π J 2 ( α ) cos 2 θ .
B x x ( Δ o , τ ) = 1 2 B c c ( Δ o , τ ) Ω ( Δ o , τ ) ,
Ω ( Δ o , τ ) 2 π 2 L 0 1 d ζ 0 d K K 3 ζ 2 Φ ( K ) β 2 ( K ζ D 2 ) J 2 ( K ( 1 ζ ) Δ o U τ ) cos ( 2 θ ) cos 2 [ K 2 μ 2 ζ ( 1 ζ ) 4 π ] .
B y y ( Δ o , τ ) = 1 2 B c c ( Δ o , τ ) + Ω ( Δ o , τ ) .
I ( α , θ ) = 0 2 π d ϕ cos ( ϕ ) sin ( ϕ ) exp [ i α cos ( ϕ θ ) ] = π J 2 ( α ) sin 2 θ ,
B x y ( Δ o , τ ) 2 π 2 L 0 1 d ζ 0 d K K 3 ζ 2 Φ ( K ) β 2 ( K ζ D 2 ) J 2 ( K ( 1 ζ ) Δ o U τ ) sin ( 2 θ ) cos 2 [ K 2 μ 2 ζ ( 1 ζ ) 4 π ] .
B x h ( Δ o , τ ) 4 π 2 k L 0 1 d ζ 0 d K K 2 Φ ( K ) β 2 ( K ζ D 2 ) ζ cos ( θ ) × J 1 ( K ( 1 ζ ) Δ o U τ ) sin [ K 2 μ 2 ζ ( 1 ζ ) 2 π ] ,
B y h ( Δ o , τ ) 4 π 2 k L 0 1 d ζ 0 d K K 2 Φ ( K ) β 2 ( K ζ D 2 ) ζ sin ( θ ) × J 1 ( K ( 1 ζ ) Δ o U τ ) sin [ K 2 μ 2 ζ ( 1 ζ ) 2 π ] .
Φ ( K ) = 0.033 C n 2 K 11 3 .
Φ ( K , L o ) = 0.033 C n 2 ( K o 2 + K 2 ) 11 6 ,
B h h ( Δ o , τ ) 48.6 σ I 2 0 1 d ζ 0 d K * K * ( K o 2 μ 2 + K * 2 ) 11 6 β 2 ( K * ζ D * 2 ) β 2 ( K * ( 1 ζ ) a * 2 ) J 0 ( K * ( 1 ζ ) ( Δ o x μ ) f c τ ) sin 2 [ K * 2 ζ ( 1 ζ ) 4 π ] ,
B c c ( Δ o , τ ) 0.448 δ c 2 0 1 d ζ 0 d K * K * 3 ( K o 2 D 2 + K * 2 ) 11 6 β 2 ( K * ζ 2 ) ζ 2 β 2 ( K * ( 1 ζ ) a * 2 ) J 0 ( K * ( 1 ζ ) ( Δ o x D ) f c τ ) cos 2 [ K * 2 μ 2 ζ ( 1 ζ ) 4 π D 2 ] ,
B h h ( Δ o , τ ) = 0 1 d ζ Q h h ( ζ , Δ o , τ ) ,

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