Abstract

The Rytov theory is applied to find the wave structure function of a laser beam transmitted from one satellite to another and propagating through the turbulent atmosphere. The phase-screen approximation is used. Taking into account refractive-index anisotropy, outer scale, and atmospheric mean-refraction defocusing, we provide expressions of the wave structure function for a spherical wave. The width and time of coherence at the receiver are evaluated. Expression for the beam spread is found using the extended Huygens–Fresnel principle, and beam wander is assessed. Beam wander occurs only for very narrow beams. Links involving low-Earth-orbit and geosynchronous satellites are studied as examples. Finally, conditions where optical tracking is perturbed by the atmosphere are examined.

© 2009 Optical Society of America

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References

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  1. W. B. Hubbard, J. R. Jokipii, and B. A. Wilking, “Stellar occultation by turbulent planetary atmospheres: a wave-optical theory including a finite scale height,” Icarus 34, 374-395(1978).
    [CrossRef]
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    [CrossRef]
  3. R. Woo, A. Ishimaru, and F.-Ch. Yang, “Radio scintillations during occultations by turbulent planetary atmospheres,” Radio Sci. 15, 695-703 (1980).
    [CrossRef]
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    [CrossRef]
  5. R. Woo, “Spacecraft radio scintillation and solar system exploration,” in Wave Propagation in Random Media (Scintillation), V. I. Tatarskii, A. Ishimaru, and V. U. Zavorotny, eds. (SPIE Press, 1993), pp. 50-83.
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    [CrossRef]
  7. A. S. Gurvich, V.Kan, and S. A. Savchenko, “Studying the turbulence and internal waves in the stratosphere from spacecraft observations of stellar scintillation: II. Probability distributions and scintillation spectra,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 37, 452-465 (2001).
  8. A. S. Gurvich, V. F. Sofieva, and F. Dalaudier, “Global distribution of CT2 at altitudes 30-50 km from space-borne observations of stellar scintillation,” Geophys. Res. Lett. 34, L24813 (2007).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  29. R. L. Lucke and C. Y. Young, “Theoretical wave structure function when the effect of the outer scale is significant,” Appl. Opt. 46, 559-569 (2007).
    [CrossRef] [PubMed]
  30. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).
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    [PubMed]

2008 (2)

2007 (4)

R. L. Lucke and C. Y. Young, “Theoretical wave structure function when the effect of the outer scale is significant,” Appl. Opt. 46, 559-569 (2007).
[CrossRef] [PubMed]

Y. Cheon and A. Muschinski, “Closed-form approximations for the angle-of-arrival variance of plane and spherical waves propagating through homogeneous and isotropic turbulence,” J. Opt. Soc. Am. A 24, 3478-3492 (2007).
[CrossRef]

A. S. Gurvich, V. F. Sofieva, and F. Dalaudier, “Global distribution of CT2 at altitudes 30-50 km from space-borne observations of stellar scintillation,” Geophys. Res. Lett. 34, L24813 (2007).
[CrossRef]

Y. Takayama, T. Jono, Y. Koyama, N. Kura, K. Shiratama, B. Demelenne, Z. Sodnik, A. Bird, and K. Arai, “Observation of atmospheric influence on OICETS inter-orbit laser communication demonstrations,” Proc. SPIE 6709, 67091B (2007).
[CrossRef]

2006 (2)

L. C. Andrews, R. L. Phillips, R. J. Sasiela, and R. R. Parenti, “Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects,” Opt. Eng. 45, 076001 (2006).
[CrossRef]

Y. Cai and S. He, “Average intensity and spreading of an elliptical Gaussian beam propagating in a turbulent atmosphere,” Opt. Lett. 31, 568-570 (2006).
[CrossRef] [PubMed]

2003 (1)

A. S. Gurvich and I. P. Chunchuzov, “Parameters of the fine density structure in the stratosphere obtained from spacecraft observations of stellar scintillations,” J. Geophys. Res. 108(D5), 4166 (2003).
[CrossRef]

2001 (3)

A. S. Gurvich and V. L. Brekhovskikh, “Study of the turbulence and inner waves in the stratosphere based on the observations of stellar scintillations from space: a model of scintillation spectra,” Waves Random Media 11, 163-181 (2001).
[CrossRef]

A. S. Gurvich, V.Kan, and S. A. Savchenko, “Studying the turbulence and internal waves in the stratosphere from spacecraft observations of stellar scintillation: II. Probability distributions and scintillation spectra,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 37, 452-465 (2001).

F. Dalaudier, V. Kan, and A. S. Gurvich, “Chromatic refraction with global ozone monitoring by occultation of stars. I. Description and scintillation correction,” Appl. Opt. 40, 866-877(2001).
[CrossRef]

1995 (2)

1992 (1)

1988 (1)

R. Narayan and W. B. Hubbard, “Theory of anisotropic refractive scintillation: application to stellar occultations by Neptune,” Astrophys. J. 325, 503-518 (1988).
[CrossRef]

1980 (1)

R. Woo, A. Ishimaru, and F.-Ch. Yang, “Radio scintillations during occultations by turbulent planetary atmospheres,” Radio Sci. 15, 695-703 (1980).
[CrossRef]

1979 (2)

B. S. Haugstad, “Turbulence in planetary occultations, IV. Power spectra of phase and intensity fluctuations,” Icarus 37, 322-335 (1979).
[CrossRef]

G. C. Loos and C. B. Hogge, “Turbulence of the upper atmosphere and isoplanatism,” Appl. Opt. 18, 2654-2661 (1979).
[CrossRef] [PubMed]

1978 (1)

W. B. Hubbard, J. R. Jokipii, and B. A. Wilking, “Stellar occultation by turbulent planetary atmospheres: a wave-optical theory including a finite scale height,” Icarus 34, 374-395(1978).
[CrossRef]

1977 (1)

1972 (1)

1971 (1)

1965 (1)

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

Andrews, L.

L. Andrews, R. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE Press, 2005).
[CrossRef]

Andrews, L. C.

L. C. Andrews, R. L. Phillips, R. J. Sasiela, and R. R. Parenti, “Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects,” Opt. Eng. 45, 076001 (2006).
[CrossRef]

Arai, K.

Y. Takayama, T. Jono, Y. Koyama, N. Kura, K. Shiratama, B. Demelenne, Z. Sodnik, A. Bird, and K. Arai, “Observation of atmospheric influence on OICETS inter-orbit laser communication demonstrations,” Proc. SPIE 6709, 67091B (2007).
[CrossRef]

Basu, S.

Belen'kii, M. S.

Bird, A.

Y. Takayama, T. Jono, Y. Koyama, N. Kura, K. Shiratama, B. Demelenne, Z. Sodnik, A. Bird, and K. Arai, “Observation of atmospheric influence on OICETS inter-orbit laser communication demonstrations,” Proc. SPIE 6709, 67091B (2007).
[CrossRef]

Brekhovskikh, V. L.

A. S. Gurvich and V. L. Brekhovskikh, “Study of the turbulence and inner waves in the stratosphere based on the observations of stellar scintillations from space: a model of scintillation spectra,” Waves Random Media 11, 163-181 (2001).
[CrossRef]

Cai, Y.

Cheon, Y.

Chunchuzov, I. P.

A. S. Gurvich and I. P. Chunchuzov, “Parameters of the fine density structure in the stratosphere obtained from spacecraft observations of stellar scintillations,” J. Geophys. Res. 108(D5), 4166 (2003).
[CrossRef]

Conan, J.-M.

Dalaudier, F.

Demelenne, B.

Y. Takayama, T. Jono, Y. Koyama, N. Kura, K. Shiratama, B. Demelenne, Z. Sodnik, A. Bird, and K. Arai, “Observation of atmospheric influence on OICETS inter-orbit laser communication demonstrations,” Proc. SPIE 6709, 67091B (2007).
[CrossRef]

Fried, D. L.

Greenwood, D. P.

Gurvich, A. S.

A. S. Gurvich, V. F. Sofieva, and F. Dalaudier, “Global distribution of CT2 at altitudes 30-50 km from space-borne observations of stellar scintillation,” Geophys. Res. Lett. 34, L24813 (2007).
[CrossRef]

A. S. Gurvich and I. P. Chunchuzov, “Parameters of the fine density structure in the stratosphere obtained from spacecraft observations of stellar scintillations,” J. Geophys. Res. 108(D5), 4166 (2003).
[CrossRef]

F. Dalaudier, V. Kan, and A. S. Gurvich, “Chromatic refraction with global ozone monitoring by occultation of stars. I. Description and scintillation correction,” Appl. Opt. 40, 866-877(2001).
[CrossRef]

A. S. Gurvich, V.Kan, and S. A. Savchenko, “Studying the turbulence and internal waves in the stratosphere from spacecraft observations of stellar scintillation: II. Probability distributions and scintillation spectra,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 37, 452-465 (2001).

A. S. Gurvich and V. L. Brekhovskikh, “Study of the turbulence and inner waves in the stratosphere based on the observations of stellar scintillations from space: a model of scintillation spectra,” Waves Random Media 11, 163-181 (2001).
[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]

Hardy, J. W.

J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford University Press, 1998).

Haugstad, B. S.

B. S. Haugstad, “Turbulence in planetary occultations, IV. Power spectra of phase and intensity fluctuations,” Icarus 37, 322-335 (1979).
[CrossRef]

He, S.

Hogge, C. B.

Hubbard, W. B.

R. Narayan and W. B. Hubbard, “Theory of anisotropic refractive scintillation: application to stellar occultations by Neptune,” Astrophys. J. 325, 503-518 (1988).
[CrossRef]

W. B. Hubbard, J. R. Jokipii, and B. A. Wilking, “Stellar occultation by turbulent planetary atmospheres: a wave-optical theory including a finite scale height,” Icarus 34, 374-395(1978).
[CrossRef]

Ishimaru, A.

R. Woo, A. Ishimaru, and F.-Ch. Yang, “Radio scintillations during occultations by turbulent planetary atmospheres,” Radio Sci. 15, 695-703 (1980).
[CrossRef]

Jokipii, J. R.

W. B. Hubbard, J. R. Jokipii, and B. A. Wilking, “Stellar occultation by turbulent planetary atmospheres: a wave-optical theory including a finite scale height,” Icarus 34, 374-395(1978).
[CrossRef]

Jono, T.

Y. Takayama, T. Jono, Y. Koyama, N. Kura, K. Shiratama, B. Demelenne, Z. Sodnik, A. Bird, and K. Arai, “Observation of atmospheric influence on OICETS inter-orbit laser communication demonstrations,” Proc. SPIE 6709, 67091B (2007).
[CrossRef]

Kan, V.

F. Dalaudier, V. Kan, and A. S. Gurvich, “Chromatic refraction with global ozone monitoring by occultation of stars. I. Description and scintillation correction,” Appl. Opt. 40, 866-877(2001).
[CrossRef]

A. S. Gurvich, V.Kan, and S. A. Savchenko, “Studying the turbulence and internal waves in the stratosphere from spacecraft observations of stellar scintillation: II. Probability distributions and scintillation spectra,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 37, 452-465 (2001).

Koyama, Y.

Y. Takayama, T. Jono, Y. Koyama, N. Kura, K. Shiratama, B. Demelenne, Z. Sodnik, A. Bird, and K. Arai, “Observation of atmospheric influence on OICETS inter-orbit laser communication demonstrations,” Proc. SPIE 6709, 67091B (2007).
[CrossRef]

Kura, N.

Y. Takayama, T. Jono, Y. Koyama, N. Kura, K. Shiratama, B. Demelenne, Z. Sodnik, A. Bird, and K. Arai, “Observation of atmospheric influence on OICETS inter-orbit laser communication demonstrations,” Proc. SPIE 6709, 67091B (2007).
[CrossRef]

Loos, G. C.

Lucke, R. L.

Lutomorski, R. F.

Michau, V.

Muschinski, A.

Narayan, R.

R. Narayan and W. B. Hubbard, “Theory of anisotropic refractive scintillation: application to stellar occultations by Neptune,” Astrophys. J. 325, 503-518 (1988).
[CrossRef]

Parenti, R. R.

L. C. Andrews, R. L. Phillips, R. J. Sasiela, and R. R. Parenti, “Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects,” Opt. Eng. 45, 076001 (2006).
[CrossRef]

Phillips, R.

L. Andrews, R. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE Press, 2005).
[CrossRef]

Phillips, R. L.

L. C. Andrews, R. L. Phillips, R. J. Sasiela, and R. R. Parenti, “Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects,” Opt. Eng. 45, 076001 (2006).
[CrossRef]

Renard, J.-B.

Robert, C.

Sasiela, R. J.

L. C. Andrews, R. L. Phillips, R. J. Sasiela, and R. R. Parenti, “Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects,” Opt. Eng. 45, 076001 (2006).
[CrossRef]

R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence; Evaluation and Application of Mellin Transforms, 2nd ed. (SPIE Press, 2007).
[PubMed]

Savchenko, S. A.

A. S. Gurvich, V.Kan, and S. A. Savchenko, “Studying the turbulence and internal waves in the stratosphere from spacecraft observations of stellar scintillation: II. Probability distributions and scintillation spectra,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 37, 452-465 (2001).

Shiratama, K.

Y. Takayama, T. Jono, Y. Koyama, N. Kura, K. Shiratama, B. Demelenne, Z. Sodnik, A. Bird, and K. Arai, “Observation of atmospheric influence on OICETS inter-orbit laser communication demonstrations,” Proc. SPIE 6709, 67091B (2007).
[CrossRef]

Sodnik, Z.

Y. Takayama, T. Jono, Y. Koyama, N. Kura, K. Shiratama, B. Demelenne, Z. Sodnik, A. Bird, and K. Arai, “Observation of atmospheric influence on OICETS inter-orbit laser communication demonstrations,” Proc. SPIE 6709, 67091B (2007).
[CrossRef]

Sofieva, V. F.

A. S. Gurvich, V. F. Sofieva, and F. Dalaudier, “Global distribution of CT2 at altitudes 30-50 km from space-borne observations of stellar scintillation,” Geophys. Res. Lett. 34, L24813 (2007).
[CrossRef]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

Takayama, Y.

Y. Takayama, T. Jono, Y. Koyama, N. Kura, K. Shiratama, B. Demelenne, Z. Sodnik, A. Bird, and K. Arai, “Observation of atmospheric influence on OICETS inter-orbit laser communication demonstrations,” Proc. SPIE 6709, 67091B (2007).
[CrossRef]

Y. Takayama, National Institute of Information and Communications Technology (personal communication, 2008).

Tatarskii, V. I.

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

Tofsted, D. H.

Voelz, D.

Wilking, B. A.

W. B. Hubbard, J. R. Jokipii, and B. A. Wilking, “Stellar occultation by turbulent planetary atmospheres: a wave-optical theory including a finite scale height,” Icarus 34, 374-395(1978).
[CrossRef]

Woo, R.

R. Woo, A. Ishimaru, and F.-Ch. Yang, “Radio scintillations during occultations by turbulent planetary atmospheres,” Radio Sci. 15, 695-703 (1980).
[CrossRef]

R. Woo, “Spacecraft radio scintillation and solar system exploration,” in Wave Propagation in Random Media (Scintillation), V. I. Tatarskii, A. Ishimaru, and V. U. Zavorotny, eds. (SPIE Press, 1993), pp. 50-83.

Yang, F.-Ch.

R. Woo, A. Ishimaru, and F.-Ch. Yang, “Radio scintillations during occultations by turbulent planetary atmospheres,” Radio Sci. 15, 695-703 (1980).
[CrossRef]

Young, C. Y.

Yura, H. T.

Appl. Opt. (5)

Astrophys. J. (1)

R. Narayan and W. B. Hubbard, “Theory of anisotropic refractive scintillation: application to stellar occultations by Neptune,” Astrophys. J. 325, 503-518 (1988).
[CrossRef]

Geophys. Res. Lett. (1)

A. S. Gurvich, V. F. Sofieva, and F. Dalaudier, “Global distribution of CT2 at altitudes 30-50 km from space-borne observations of stellar scintillation,” Geophys. Res. Lett. 34, L24813 (2007).
[CrossRef]

Icarus (2)

W. B. Hubbard, J. R. Jokipii, and B. A. Wilking, “Stellar occultation by turbulent planetary atmospheres: a wave-optical theory including a finite scale height,” Icarus 34, 374-395(1978).
[CrossRef]

B. S. Haugstad, “Turbulence in planetary occultations, IV. Power spectra of phase and intensity fluctuations,” Icarus 37, 322-335 (1979).
[CrossRef]

Izv. Acad. Sci. USSR Atmos. Oceanic Phys. (1)

A. S. Gurvich, V.Kan, and S. A. Savchenko, “Studying the turbulence and internal waves in the stratosphere from spacecraft observations of stellar scintillation: II. Probability distributions and scintillation spectra,” Izv. Acad. Sci. USSR Atmos. Oceanic Phys. 37, 452-465 (2001).

J. Geophys. Res. (1)

A. S. Gurvich and I. P. Chunchuzov, “Parameters of the fine density structure in the stratosphere obtained from spacecraft observations of stellar scintillations,” J. Geophys. Res. 108(D5), 4166 (2003).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (4)

Opt. Eng. (1)

L. C. Andrews, R. L. Phillips, R. J. Sasiela, and R. R. Parenti, “Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects,” Opt. Eng. 45, 076001 (2006).
[CrossRef]

Opt. Lett. (2)

Proc. SPIE (1)

Y. Takayama, T. Jono, Y. Koyama, N. Kura, K. Shiratama, B. Demelenne, Z. Sodnik, A. Bird, and K. Arai, “Observation of atmospheric influence on OICETS inter-orbit laser communication demonstrations,” Proc. SPIE 6709, 67091B (2007).
[CrossRef]

Radio Sci. (1)

R. Woo, A. Ishimaru, and F.-Ch. Yang, “Radio scintillations during occultations by turbulent planetary atmospheres,” Radio Sci. 15, 695-703 (1980).
[CrossRef]

Waves Random Media (1)

A. S. Gurvich and V. L. Brekhovskikh, “Study of the turbulence and inner waves in the stratosphere based on the observations of stellar scintillations from space: a model of scintillation spectra,” Waves Random Media 11, 163-181 (2001).
[CrossRef]

Other (7)

R. Woo, “Spacecraft radio scintillation and solar system exploration,” in Wave Propagation in Random Media (Scintillation), V. I. Tatarskii, A. Ishimaru, and V. U. Zavorotny, eds. (SPIE Press, 1993), pp. 50-83.

J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford University Press, 1998).

L. Andrews, R. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE Press, 2005).
[CrossRef]

Y. Takayama, National Institute of Information and Communications Technology (personal communication, 2008).

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

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence; Evaluation and Application of Mellin Transforms, 2nd ed. (SPIE Press, 2007).
[PubMed]

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Figures (13)

Fig. 1
Fig. 1

Considered atmospheric profiles of turbulence: C n 2 ( h ) and S n 2 ( h ) .

Fig. 2
Fig. 2

Considered link geometry.

Fig. 3
Fig. 3

Beam coordinate system and α parameter.

Fig. 4
Fig. 4

Geometry of the considered beam model and angular relations (PS, phase screen).

Fig. 5
Fig. 5

Vertical expansion factor ϕ 1 L / L 1 for four different links.

Fig. 6
Fig. 6

Normalized coherence widths ρ 0 , atm , x and ρ 0 , atm , y defined with L 2 = 0 and plotted for λ = 1 μm . The corresponding values for the diffraction angles θ atm , x and θ atm , y are displayed on the right axis.

Fig. 7
Fig. 7

Fluctuations of the AoA as recorded by the OICETS satellite during a link with the ARTEMIS satellite. The upper graph shows the vertical angular component and the lower graph shows the horizontal one (courtesy of Takayama et al.. [9]).

Fig. 8
Fig. 8

Transverse component of the relative wind for a “LEO-chasing-LEO” link.

Fig. 9
Fig. 9

Coherence time τ 0 calculated as a function of H P for various scenarios.

Fig. 10
Fig. 10

Geometry of an occultation link with a LEO satellite and a remote countersatellite: (a) the LEO orbit makes an angle α orbit with respect to the direction toward the countersatellite, (b) the LEO orbit is traced as seen from the countersatellite.

Fig. 11
Fig. 11

Relative transverse wind V RW as a function of the LEO orbit inclination α orbit in the case of a LEO–GEO link.

Fig. 12
Fig. 12

Beam-spread loss as a function of perigee altitude for λ = 1 μm .

Fig. 13
Fig. 13

Relative beam wander β x and β y for a LEO-to-GEO link with θ div = 3 μrad and λ = 1 μm .

Tables (1)

Tables Icon

Table 1 Geometrical Parameters Relevant for Links with LEO and/or GEO Satellites

Equations (73)

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

Φ n ( κ ) = Φ n , i ( κ ) + Φ n , a ( κ ) .
Φ n , i ( κ ) = 0.033 C n 2 ( κ 2 + κ 0 , i 2 ) 11 / 6 ,
C n 2 ( h ) = HV 5 / 7 ( h ) + N ( h ) 2 C K ( h ) .
C K ( h ) = 10 10 e h / 10 km [ m 2 / 3 ] .
Φ n , a ( K ) = S n 2 ( K 2 + κ 0 , a 2 ) 5 / 2 ,
K 2 = η 2 ( K X 2 + K Y 2 ) + K Z 2 .
S n 2 = η 2 N 2 C W .
C W ( h ) = 5 × 10 11 1 1 + 4.5 × 10 8 N 2 [ m 2 ] .
L 0 , a ( h ) e h / 10 km [ m ] .
μ 0 , i 0 L C n 2 [ h ( z ) ] d z .
L 1 L atm L 1 + L atm C n 2 [ h ( z ) ] d z = p μ 0 , i .
L atm erfinv ( p ) H 0 R e ,
L R L 1 ,
θ div W 1 L 1 W L .
1 k L 2 W 1 ρ W .
1 k L 2 H 0 ρ x .
ϕ 1 = 1 L 1 L 2 L d ω d h .
D w ( ρ x , ρ y ) = 0 L D w z ( ρ x , ρ y ; z ) d z ,
D w z ( ρ x , ρ y ; z ) = 4 π k 2 Φ n ( κ x , κ y , z ) [ 1 e j ξ ( ϕ κ x ρ x + κ y ρ y ) ] d κ x d κ y ,
D w , i z ( ρ x , ρ y ; z ) = 8 π 2 k 2 0 Φ n , i ( κ ; z ) [ 1 J 0 ( κ r i ) ] κ d κ ,
r i = L 1 L ( ϕ ρ x ) 2 + ρ y 2 .
D w , i ( ρ x , ρ y ) = 2 ( r i ρ i , atm ) 5 / 3 , r i L 0 , i ,
D w , i ( ρ x , ρ y ) = 0.05 ( L 0 , i ρ i , atm ) 5 / 3 , r i L 0 , i .
Φ n , a ( κ x , κ y , z ) = S n 2 ( η 2 K X 2 + K Z 2 + κ 0 , a 2 ) 5 / 2 = S n 2 ( η 2 κ y 2 + γ 2 κ x 2 + κ 0 , a 2 ) 5 / 2 ,
γ = cos 2 α + η 2 sin 2 α .
D w , a z ( ρ x , ρ y ; z ) = 8 π 2 k 2 S n 2 ( γ η ) 1 0 1 J 0 ( κ r a ) ( κ 2 + κ 0 , a 2 ) 5 / 2 κ d κ ,
r a = L 1 L ( ϕ γ 1 ρ x ) 2 + ( η 1 ρ y ) 2 .
μ m , a 0 L S n 2 γ m d z ,
D w , a ( ρ x , ρ y ) = 2 ( r a ρ a , atm ) 2 , r a L 0 , a ,
D w , a ( ρ x , ρ y ) = 1 6 π 2 ( L 0 , a ρ a , atm ) 2 , r a L 0 , a .
D w ( ρ 0 , x , 0 ) D w ( 0 , ρ 0 , y ) 2.
D w ( ρ 0 , atm , x , 0 ) | L 2 = 0 D w ( 0 , ρ 0 , atm , y ) | L 2 = 0 2.
[ ρ 0 , x ρ 0 , y ] L L 1 [ ϕ 1 ρ 0 , atm , x ρ 0 , atm , y ] .
[ θ atm , x θ atm , y ] 2 k [ 1 / ρ 0 , atm , x 1 / ρ 0 , atm , y ] .
D w ( 0 , ρ y ) D w , i ( 0 , ρ y ) .
[ σ AoA , x 2 σ AoA , y 2 ] = 1 k 2 D 2 [ D w ( D , 0 ) D w ( 0 , D ) ] .
[ σ AoA , x σ AoA , y ] 2 k [ 1 / ρ 0 , x 1 / ρ 0 , y ] .
[ σ AoA , x σ AoA , y ] 1 2 L 1 L [ ϕ θ atm , x θ atm , y ] .
V RW , 0 ( z ) = ( 1 ξ ) V 1 , + ξ V 2 , .
[ V RW , x V RW , y ] = [ ϕ V RW , 0 , x V RW , 0 , y ] , | z L 1 | < L atm .
[ V RW , x V RW , y ] τ = ξ [ ϕ ρ x ρ y ] .
D w , i z ( τ ; z ) = 2.9 k 2 C n 2 ( V RW τ ) 5 / 3 ,
D w , a z ( τ ; z ) = 1 / 2 π k 2 L 0 , a S n 2 ( γ η ) 1 ( γ 2 V RW , x 2 + η 2 V RW , y 2 ) τ 2 .
D w , i ( τ ) = 2 ( V eq , i τ ρ i , atm ) 5 / 3 ,
D w , a ( τ ) = 2 ( V eq , a , x 2 + η 2 V eq , a , y 2 τ ρ a , atm ) 2 ,
V eq , i = ( μ 0 , i 1 0 L C n 2 V RW 5 / 3 d z ) 3 / 5 ,
V eq , a , x = ( μ 3 , a 1 0 L S n 2 γ 3 V RW , x 2 d z ) 1 / 2 ,
V eq , a , y = ( μ 1 , a 1 0 L S n 2 γ 1 V RW , y 2 d z ) 1 / 2 .
τ 0 = { ρ 0 , atm V eq , i if   D S , a ( τ ) D S , i ( τ ) ( V eq , a , x 2 ρ 0 , atm , x 2 + V eq , a , y 2 ρ 0 , atm , y 2 ) - 1 / 2 if   D S , i ( τ ) D S , a ( τ ) .
V 1 , = V 1 [ L 1 / ( R e + H 1 ) 0 ] V 2 , = V 1 , .
[ V RW , x V RW , y ] = [ ϕ ( 1 2 ξ ) V 1 , x 0 ] .
[ V RW , x V RW , y ] [ ϕ LEO V LEO , x V LEO , y ] .
I ( x , y ) = k 2 W 2 8 π L 2 2 d Q x d Q y exp [ i k L 2 ( x Q x + y Q y ) ] × exp [ k 2 W 2 8 L 2 2 ( Q x 2 ϕ 2 + Q y 2 ) ] exp [ 1 2 D sp ( Q x , Q y ) ] .
D sp ( Q x , Q y ) = 2 ( Q x 2 ρ 0 , atm , x 2 + Q y 2 ρ 0 , atm , y 2 ) .
I ( x , y ) = 1 s x s y exp [ 2 x 2 s x 2 W 2 2 y 2 s y 2 W 2 ] ,
s x 2 = ϕ 2 + 2 ( L 2 L θ atm , x θ div ) 2 ,
s y 2 = 1 + 2 ( L 2 L θ atm , y θ div ) 2 .
[ σ BW , x 2 σ BW , y 2 ] = 1 4 k 2 W 1 2 [ D w ( 2 W 1 , 0 ) D w ( 0 , 2 W 1 ) ] | L 2 = 0 ,
[ σ BW , x σ BW , y ] 1 2 [ θ atm , x θ atm , y ] , W 1 L 0 ,
[ σ BW , x σ BW , y ] c L 0 W 1 [ θ atm , x θ atm , y ] , W 1 L 0 ,
W LT , x 2 = W ST , x 2 + 2 σ c , x 2 ,
β x 2 2 σ c , x 2 W ST , x 2 = 1 W LT , x 2 / ( 2 σ c , x 2 ) 1 .
β x [ ( L ϕ L 2 θ div θ atm , x ) 2 + 1 ] 1 / 2 if     W 1 L 0 ,
β x c L 0 W 1 [ ( L ϕ L 2 θ div θ atm , x ) 2 + 2 ] 1 / 2 if     W 1 c L 0 .
D ga ( ρ ) D sp ( ρ ) = Γ ( 11 / 6 ) [ k ( 1 d 3 ) 2 ρ 2 4 Λ L d 3 2 ] 5 / 6 × { F 1 1 [ 5 6 ; 1 ; k ( 1 Θ ¯ d 3 ) 2 ρ 2 4 Λ L d 3 2 ] F 1 1 [ 5 6 ; 1 ; k Λ ρ 2 4 L ] } ,
d 3 = L 2 / L Θ ¯ 1 Λ = 2 L / ( k W 2 ) .
k L 1 2 ρ 2 4 Λ L L 2 2 1 k Λ ρ 2 4 L 1 ,
I p = 0 1 J 0 ( κ ρ ) ( κ 2 + κ 0 2 ) p κ d κ ,
I p = κ 0 2 ( 1 p ) 2 ( p 1 ) [ 1 2 Γ ( p 1 ) ( κ 0 ρ 2 ) p 1 K p 1 ( κ 0 ρ ) ] ,
I p κ 0 2 ( 1 p ) 2 ( p 1 ) , for     κ 0 ρ .
2 Γ ( v ) ( x 2 ) v K v ( x ) = n = 0 ( 1 ) n n ! [ Γ ( n v ) Γ ( v ) ( x 2 ) 2 n + 2 v + Γ ( n + v ) Γ ( v ) ( x 2 ) 2 n ] .
I 11 / 6 3 5 Γ ( 1 / 6 ) Γ ( 11 / 6 ) ( ρ 2 ) 5 / 3 , κ 0 ρ 1 I 11 / 6 3 5 1 κ 0 5 / 3 , κ 0 ρ 1 ,
I 5 / 2 ρ 2 8 κ 0 , κ 0 ρ 1 I 5 / 2 1 3 κ 0 3 , κ 0 ρ 1 .

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