Abstract

An approximation is derived for the phase Strehl of an aberrated wavefront based on uncorrelated random variates. Eliminating the requirement to generate correlated variates offers an orders-of-magnitude improvement in simulation speed, while yielding accuracy that may be sufficient for the preliminary sizing of adaptive optics (AO) in laser communications system design. Examples are presented comparing the performance of several AO subsystem sizes when correcting a wavefront aberrated by Kolmogorov turbulence.

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References

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  1. M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40, 1554–1562 (2001).
    [CrossRef]
  2. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Theory of optical scintillation: Gaussian-beam wave model,” in Eighth Joint International Symposium on Atmospheric and Ocean Optics: Atmospheric Physics (International Society for Optics and Photonics, 2002), pp. 91–101.
  3. L. C. Andrews and R. L. Phillips, “IK distribution as a universal propagation model of laser beams in atmospheric turbulence,” J. Opt. Soc. Am. A 2, 160–163 (1985).
    [CrossRef]
  4. L. C. Andrews and R. L. Phillips, “Mathematical genesis of the IK distribution for random optical fields,” J. Opt. Soc. Am. A 3, 1912–1919 (1986).
    [CrossRef]
  5. D. K. Borah and D. G. Voelz, “Pointing error effects on free-space optical communication links in the presence of atmospheric turbulence,” J. Lightwave Technol. 27, 3965–3973 (2009).
    [CrossRef]
  6. J. H. Churnside and R. J. Hill, “Probability density of irradiance scintillations for strong path-integrated refractive turbulence,” J. Opt. Soc. Am. A 4, 727–733 (1987).
    [CrossRef]
  7. J. H. Churnside and S. F. Clifford, “Log-normal Rician probability-density function of optical scintillations in the turbulent atmosphere,” J. Opt. Soc. Am. A 4, 1923–1930 (1987).
    [CrossRef]
  8. J. H. Churnside and R. G. Frehlich, “Experimental evaluation of log-normally modulated Rician and IK models of optical scintillation in the atmosphere,” J. Opt. Soc. Am. A 6, 1760–1766 (1989).
    [CrossRef]
  9. J. H. Churnside, “Aperture averaging of optical scintillations in the turbulent atmosphere,” Appl. Opt. 30, 1982–1994 (1991).
    [CrossRef]
  10. R. Dashen, G.-Y. Wang, S. M. Flatte, and C. Bracher, “Moments of intensity and log intensity: new asymptotic results for waves in power-law media,” J. Opt. Soc. Am. A 10, 1233–1242 (1993).
    [CrossRef]
  11. F. M. Davidson, G. C. Gilbreath, and E. Oh, “Measurements of intensity scintillations and probability density functions of retroreflected broadband 980-nm laser light in atmospheric turbulence,” Opt. Eng. 43, 2689–2695 (2004).
    [CrossRef]
  12. S. M. Flatté, C. Bracher, and G.-Y. Wang, “Probability-density functions of irradiance for waves in atmospheric turbulence calculated by numerical simulation,” J. Opt. Soc. Am. A 11, 2080–2092 (1994).
    [CrossRef]
  13. R. J. Hill and R. G. Frehlich, “Probability distribution of irradiance for the onset of strong scintillation,” J. Opt. Soc. Am. A 14, 1530–1540 (1997).
    [CrossRef]
  14. K. Kiasaleh, “On the probability density function of signal intensity in free-space optical communications systems impaired by pointing jitter and turbulence,” Opt. Eng. 33, 3748–3757 (1994).
    [CrossRef]
  15. S. D. Lyke, D. G. Voelz, and M. C. Roggemann, “Probability density of aperture-averaged irradiance fluctuations for long range free space optical communication links,” Appl. Opt. 48, 6511–6527 (2009).
    [CrossRef]
  16. G. Parry and P. N. Pusey, “K distributions in atmospheric propagation of laser light,” J. Opt. Soc. Am. 69, 796–798 (1979).
    [CrossRef]
  17. R. L. Phillips and L. C. Andrews, “Universal statistical model for irradiance fluctuations in a turbulent medium,” J. Opt. Soc. Am. 72, 864–870 (1982).
    [CrossRef]
  18. M. C. Teich and P. Diament, “Multiply stochastic representations for K distributions and their Poisson transforms,” J. Opt. Soc. Am. A 6, 80–91 (1989).
    [CrossRef]
  19. M. Toyoshima, S. Yamakawa, T. Yamawaki, K. Arai, M. R. García-Talavera, A. Alonso, Z. Sodnik, and B. Demelenne, “Long-term statistics of laser beam propagation in an optical ground-to-geostationary satellite communications link,” IEEE Trans. Antennas Propag. 53, 842–850 (2005).
    [CrossRef]
  20. F. S. Vetelino, C. Young, L. Andrews, and J. Recolons, “Aperture averaging effects on the probability density of irradiance fluctuations in moderate-to-strong turbulence,” Appl. Opt. 46, 2099–2108 (2007).
    [CrossRef]
  21. C. R. Ambrose, Strehl Ratio Probabilities for Phase-Only Adaptive Optics (Naval Postgraduate School, 1999).
  22. L. E. Goad, “Performance scaling laws for adaptive optics systems,” Proc. SPIE 1920, 2–8 (1993).
    [CrossRef]
  23. S. Gladyszab, J. C. Christoua, and M. Redfernb, “Characterization of the Lick adaptive optics point spread function,” Proc. SPIE 6272, 62720J (2006).
    [CrossRef]
  24. G. A. Tyler, “Assessment of the statistics of the Strehl ratio: predictions of central limit theorem analysis,” J. Opt. Soc. Am. A 23, 2834–2844 (2006).
    [CrossRef]
  25. N. Yaitskova and S. Gladysz, “First-order speckle statistics for arbitrary aberration strength,” J. Opt. Soc. Am. A 28, 1909–1919 (2011).
    [CrossRef]
  26. H. T. Yura and D. L. Fried, “Variance of the Strehl ratio of an adaptive optics system,” J. Opt. Soc. Am. A 15, 2107–2110 (1998).
    [CrossRef]
  27. D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave front,” Proc. IEEE 55, 57–77 (1967).
    [CrossRef]
  28. V. I. Tatarskii, “The effects of the turbulent atmosphere on wave propagation,” Jerusalem: Israel Program for Scientific Translations1 (1971).
  29. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
    [CrossRef]
  30. R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence: Evaluation and Application of Mellin Transforms (SPIE, 2007).

2011 (1)

2009 (2)

2007 (1)

2006 (2)

S. Gladyszab, J. C. Christoua, and M. Redfernb, “Characterization of the Lick adaptive optics point spread function,” Proc. SPIE 6272, 62720J (2006).
[CrossRef]

G. A. Tyler, “Assessment of the statistics of the Strehl ratio: predictions of central limit theorem analysis,” J. Opt. Soc. Am. A 23, 2834–2844 (2006).
[CrossRef]

2005 (1)

M. Toyoshima, S. Yamakawa, T. Yamawaki, K. Arai, M. R. García-Talavera, A. Alonso, Z. Sodnik, and B. Demelenne, “Long-term statistics of laser beam propagation in an optical ground-to-geostationary satellite communications link,” IEEE Trans. Antennas Propag. 53, 842–850 (2005).
[CrossRef]

2004 (1)

F. M. Davidson, G. C. Gilbreath, and E. Oh, “Measurements of intensity scintillations and probability density functions of retroreflected broadband 980-nm laser light in atmospheric turbulence,” Opt. Eng. 43, 2689–2695 (2004).
[CrossRef]

2001 (1)

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40, 1554–1562 (2001).
[CrossRef]

1998 (1)

1997 (1)

1994 (2)

K. Kiasaleh, “On the probability density function of signal intensity in free-space optical communications systems impaired by pointing jitter and turbulence,” Opt. Eng. 33, 3748–3757 (1994).
[CrossRef]

S. M. Flatté, C. Bracher, and G.-Y. Wang, “Probability-density functions of irradiance for waves in atmospheric turbulence calculated by numerical simulation,” J. Opt. Soc. Am. A 11, 2080–2092 (1994).
[CrossRef]

1993 (2)

1991 (1)

1989 (2)

1987 (2)

1986 (1)

1985 (1)

1982 (1)

1979 (1)

1976 (1)

1967 (1)

D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave front,” Proc. IEEE 55, 57–77 (1967).
[CrossRef]

Al-Habash, A.

A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Theory of optical scintillation: Gaussian-beam wave model,” in Eighth Joint International Symposium on Atmospheric and Ocean Optics: Atmospheric Physics (International Society for Optics and Photonics, 2002), pp. 91–101.

Al-Habash, M. A.

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40, 1554–1562 (2001).
[CrossRef]

Alonso, A.

M. Toyoshima, S. Yamakawa, T. Yamawaki, K. Arai, M. R. García-Talavera, A. Alonso, Z. Sodnik, and B. Demelenne, “Long-term statistics of laser beam propagation in an optical ground-to-geostationary satellite communications link,” IEEE Trans. Antennas Propag. 53, 842–850 (2005).
[CrossRef]

Ambrose, C. R.

C. R. Ambrose, Strehl Ratio Probabilities for Phase-Only Adaptive Optics (Naval Postgraduate School, 1999).

Andrews, L.

Andrews, L. C.

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40, 1554–1562 (2001).
[CrossRef]

L. C. Andrews and R. L. Phillips, “Mathematical genesis of the IK distribution for random optical fields,” J. Opt. Soc. Am. A 3, 1912–1919 (1986).
[CrossRef]

L. C. Andrews and R. L. Phillips, “IK distribution as a universal propagation model of laser beams in atmospheric turbulence,” J. Opt. Soc. Am. A 2, 160–163 (1985).
[CrossRef]

R. L. Phillips and L. C. Andrews, “Universal statistical model for irradiance fluctuations in a turbulent medium,” J. Opt. Soc. Am. 72, 864–870 (1982).
[CrossRef]

A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Theory of optical scintillation: Gaussian-beam wave model,” in Eighth Joint International Symposium on Atmospheric and Ocean Optics: Atmospheric Physics (International Society for Optics and Photonics, 2002), pp. 91–101.

Arai, K.

M. Toyoshima, S. Yamakawa, T. Yamawaki, K. Arai, M. R. García-Talavera, A. Alonso, Z. Sodnik, and B. Demelenne, “Long-term statistics of laser beam propagation in an optical ground-to-geostationary satellite communications link,” IEEE Trans. Antennas Propag. 53, 842–850 (2005).
[CrossRef]

Borah, D. K.

Bracher, C.

Christoua, J. C.

S. Gladyszab, J. C. Christoua, and M. Redfernb, “Characterization of the Lick adaptive optics point spread function,” Proc. SPIE 6272, 62720J (2006).
[CrossRef]

Churnside, J. H.

Clifford, S. F.

Dashen, R.

Davidson, F. M.

F. M. Davidson, G. C. Gilbreath, and E. Oh, “Measurements of intensity scintillations and probability density functions of retroreflected broadband 980-nm laser light in atmospheric turbulence,” Opt. Eng. 43, 2689–2695 (2004).
[CrossRef]

Demelenne, B.

M. Toyoshima, S. Yamakawa, T. Yamawaki, K. Arai, M. R. García-Talavera, A. Alonso, Z. Sodnik, and B. Demelenne, “Long-term statistics of laser beam propagation in an optical ground-to-geostationary satellite communications link,” IEEE Trans. Antennas Propag. 53, 842–850 (2005).
[CrossRef]

Diament, P.

Flatte, S. M.

Flatté, S. M.

Frehlich, R. G.

Fried, D. L.

H. T. Yura and D. L. Fried, “Variance of the Strehl ratio of an adaptive optics system,” J. Opt. Soc. Am. A 15, 2107–2110 (1998).
[CrossRef]

D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave front,” Proc. IEEE 55, 57–77 (1967).
[CrossRef]

García-Talavera, M. R.

M. Toyoshima, S. Yamakawa, T. Yamawaki, K. Arai, M. R. García-Talavera, A. Alonso, Z. Sodnik, and B. Demelenne, “Long-term statistics of laser beam propagation in an optical ground-to-geostationary satellite communications link,” IEEE Trans. Antennas Propag. 53, 842–850 (2005).
[CrossRef]

Gilbreath, G. C.

F. M. Davidson, G. C. Gilbreath, and E. Oh, “Measurements of intensity scintillations and probability density functions of retroreflected broadband 980-nm laser light in atmospheric turbulence,” Opt. Eng. 43, 2689–2695 (2004).
[CrossRef]

Gladysz, S.

Gladyszab, S.

S. Gladyszab, J. C. Christoua, and M. Redfernb, “Characterization of the Lick adaptive optics point spread function,” Proc. SPIE 6272, 62720J (2006).
[CrossRef]

Goad, L. E.

L. E. Goad, “Performance scaling laws for adaptive optics systems,” Proc. SPIE 1920, 2–8 (1993).
[CrossRef]

Hill, R. J.

Kiasaleh, K.

K. Kiasaleh, “On the probability density function of signal intensity in free-space optical communications systems impaired by pointing jitter and turbulence,” Opt. Eng. 33, 3748–3757 (1994).
[CrossRef]

Lyke, S. D.

Noll, R. J.

Oh, E.

F. M. Davidson, G. C. Gilbreath, and E. Oh, “Measurements of intensity scintillations and probability density functions of retroreflected broadband 980-nm laser light in atmospheric turbulence,” Opt. Eng. 43, 2689–2695 (2004).
[CrossRef]

Parry, G.

Phillips, R. L.

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40, 1554–1562 (2001).
[CrossRef]

L. C. Andrews and R. L. Phillips, “Mathematical genesis of the IK distribution for random optical fields,” J. Opt. Soc. Am. A 3, 1912–1919 (1986).
[CrossRef]

L. C. Andrews and R. L. Phillips, “IK distribution as a universal propagation model of laser beams in atmospheric turbulence,” J. Opt. Soc. Am. A 2, 160–163 (1985).
[CrossRef]

R. L. Phillips and L. C. Andrews, “Universal statistical model for irradiance fluctuations in a turbulent medium,” J. Opt. Soc. Am. 72, 864–870 (1982).
[CrossRef]

A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Theory of optical scintillation: Gaussian-beam wave model,” in Eighth Joint International Symposium on Atmospheric and Ocean Optics: Atmospheric Physics (International Society for Optics and Photonics, 2002), pp. 91–101.

Pusey, P. N.

Recolons, J.

Redfernb, M.

S. Gladyszab, J. C. Christoua, and M. Redfernb, “Characterization of the Lick adaptive optics point spread function,” Proc. SPIE 6272, 62720J (2006).
[CrossRef]

Roggemann, M. C.

Sasiela, R. J.

R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence: Evaluation and Application of Mellin Transforms (SPIE, 2007).

Sodnik, Z.

M. Toyoshima, S. Yamakawa, T. Yamawaki, K. Arai, M. R. García-Talavera, A. Alonso, Z. Sodnik, and B. Demelenne, “Long-term statistics of laser beam propagation in an optical ground-to-geostationary satellite communications link,” IEEE Trans. Antennas Propag. 53, 842–850 (2005).
[CrossRef]

Tatarskii, V. I.

V. I. Tatarskii, “The effects of the turbulent atmosphere on wave propagation,” Jerusalem: Israel Program for Scientific Translations1 (1971).

Teich, M. C.

Toyoshima, M.

M. Toyoshima, S. Yamakawa, T. Yamawaki, K. Arai, M. R. García-Talavera, A. Alonso, Z. Sodnik, and B. Demelenne, “Long-term statistics of laser beam propagation in an optical ground-to-geostationary satellite communications link,” IEEE Trans. Antennas Propag. 53, 842–850 (2005).
[CrossRef]

Tyler, G. A.

Vetelino, F. S.

Voelz, D. G.

Wang, G.-Y.

Yaitskova, N.

Yamakawa, S.

M. Toyoshima, S. Yamakawa, T. Yamawaki, K. Arai, M. R. García-Talavera, A. Alonso, Z. Sodnik, and B. Demelenne, “Long-term statistics of laser beam propagation in an optical ground-to-geostationary satellite communications link,” IEEE Trans. Antennas Propag. 53, 842–850 (2005).
[CrossRef]

Yamawaki, T.

M. Toyoshima, S. Yamakawa, T. Yamawaki, K. Arai, M. R. García-Talavera, A. Alonso, Z. Sodnik, and B. Demelenne, “Long-term statistics of laser beam propagation in an optical ground-to-geostationary satellite communications link,” IEEE Trans. Antennas Propag. 53, 842–850 (2005).
[CrossRef]

Young, C.

Yura, H. T.

Appl. Opt. (3)

IEEE Trans. Antennas Propag. (1)

M. Toyoshima, S. Yamakawa, T. Yamawaki, K. Arai, M. R. García-Talavera, A. Alonso, Z. Sodnik, and B. Demelenne, “Long-term statistics of laser beam propagation in an optical ground-to-geostationary satellite communications link,” IEEE Trans. Antennas Propag. 53, 842–850 (2005).
[CrossRef]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (12)

G. A. Tyler, “Assessment of the statistics of the Strehl ratio: predictions of central limit theorem analysis,” J. Opt. Soc. Am. A 23, 2834–2844 (2006).
[CrossRef]

N. Yaitskova and S. Gladysz, “First-order speckle statistics for arbitrary aberration strength,” J. Opt. Soc. Am. A 28, 1909–1919 (2011).
[CrossRef]

H. T. Yura and D. L. Fried, “Variance of the Strehl ratio of an adaptive optics system,” J. Opt. Soc. Am. A 15, 2107–2110 (1998).
[CrossRef]

M. C. Teich and P. Diament, “Multiply stochastic representations for K distributions and their Poisson transforms,” J. Opt. Soc. Am. A 6, 80–91 (1989).
[CrossRef]

R. Dashen, G.-Y. Wang, S. M. Flatte, and C. Bracher, “Moments of intensity and log intensity: new asymptotic results for waves in power-law media,” J. Opt. Soc. Am. A 10, 1233–1242 (1993).
[CrossRef]

S. M. Flatté, C. Bracher, and G.-Y. Wang, “Probability-density functions of irradiance for waves in atmospheric turbulence calculated by numerical simulation,” J. Opt. Soc. Am. A 11, 2080–2092 (1994).
[CrossRef]

R. J. Hill and R. G. Frehlich, “Probability distribution of irradiance for the onset of strong scintillation,” J. Opt. Soc. Am. A 14, 1530–1540 (1997).
[CrossRef]

J. H. Churnside and R. J. Hill, “Probability density of irradiance scintillations for strong path-integrated refractive turbulence,” J. Opt. Soc. Am. A 4, 727–733 (1987).
[CrossRef]

J. H. Churnside and S. F. Clifford, “Log-normal Rician probability-density function of optical scintillations in the turbulent atmosphere,” J. Opt. Soc. Am. A 4, 1923–1930 (1987).
[CrossRef]

J. H. Churnside and R. G. Frehlich, “Experimental evaluation of log-normally modulated Rician and IK models of optical scintillation in the atmosphere,” J. Opt. Soc. Am. A 6, 1760–1766 (1989).
[CrossRef]

L. C. Andrews and R. L. Phillips, “IK distribution as a universal propagation model of laser beams in atmospheric turbulence,” J. Opt. Soc. Am. A 2, 160–163 (1985).
[CrossRef]

L. C. Andrews and R. L. Phillips, “Mathematical genesis of the IK distribution for random optical fields,” J. Opt. Soc. Am. A 3, 1912–1919 (1986).
[CrossRef]

Opt. Eng. (3)

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40, 1554–1562 (2001).
[CrossRef]

K. Kiasaleh, “On the probability density function of signal intensity in free-space optical communications systems impaired by pointing jitter and turbulence,” Opt. Eng. 33, 3748–3757 (1994).
[CrossRef]

F. M. Davidson, G. C. Gilbreath, and E. Oh, “Measurements of intensity scintillations and probability density functions of retroreflected broadband 980-nm laser light in atmospheric turbulence,” Opt. Eng. 43, 2689–2695 (2004).
[CrossRef]

Proc. IEEE (1)

D. L. Fried, “Optical heterodyne detection of an atmospherically distorted signal wave front,” Proc. IEEE 55, 57–77 (1967).
[CrossRef]

Proc. SPIE (2)

L. E. Goad, “Performance scaling laws for adaptive optics systems,” Proc. SPIE 1920, 2–8 (1993).
[CrossRef]

S. Gladyszab, J. C. Christoua, and M. Redfernb, “Characterization of the Lick adaptive optics point spread function,” Proc. SPIE 6272, 62720J (2006).
[CrossRef]

Other (4)

C. R. Ambrose, Strehl Ratio Probabilities for Phase-Only Adaptive Optics (Naval Postgraduate School, 1999).

V. I. Tatarskii, “The effects of the turbulent atmosphere on wave propagation,” Jerusalem: Israel Program for Scientific Translations1 (1971).

R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence: Evaluation and Application of Mellin Transforms (SPIE, 2007).

A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Theory of optical scintillation: Gaussian-beam wave model,” in Eighth Joint International Symposium on Atmospheric and Ocean Optics: Atmospheric Physics (International Society for Optics and Photonics, 2002), pp. 91–101.

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Figures (4)

Fig. 1.
Fig. 1.

Schematic of high-data-rate lasercom receiver architecture.

Fig. 2.
Fig. 2.

Example of simulation results comparing the cdf for continuous and discrete phase Strehl for Cn2=3×1012m2/3.

Fig. 3.
Fig. 3.

Example of simulation results comparing the cdf for continuous and discrete phase Strehl for Cn2=5×1012m2/3.

Fig. 4.
Fig. 4.

Example of simulation results comparing the cdf for continuous and discrete phase Strehl for Cn2=1×1011m2/3.

Equations (19)

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Sc=|Apertureexp[iϕ(x⃗)]dx⃗Aperturedx⃗|2
Sd=|1Mm=1Mexp(iϕm)|2,
UcApertureexp[iϕ(x⃗)]dx⃗Aperturedx⃗Ud1Mm=1Mexp(iϕm).
E{Uc}=ApertureE{exp[iϕ(x⃗)]}dx⃗Aperturedx⃗=E{exp[iϕ(x⃗)]}=exp(σϕ2/2)E{Ud}=1Mm=1ME{exp(iϕm)}=E{exp(iϕ)}=exp(σ2/2),
E{exp(iϕ)}=exp(σ2/2).
E{|Uc|2}=E{|Ud|2}=1M2m=1Mn=1ME{exp[i(ϕmϕn)]}=1M+M1M(E{exp(iϕ)})2=1M+M1Mexp(σϕ2),
E{|Uc|2}=(4πD2)2ApertureApertureE{exp[i(ϕ(x⃗)ϕ(y⃗))]}dy⃗dx⃗,
E{|Uc|2}=(4πD2)22π0DrK0(r)Γexp(iϕ)(r)dr,
K0(r)=12[D2acos(rD)rD2r2].
Var{ϕ(x⃗)ϕ(y⃗)}=E{ϕ2(x⃗)}2E{ϕ(x⃗)ϕ(y⃗)}+E{ϕ2(y⃗)}=2σϕ22Γϕ(r).
Γexp(iϕ)(r)=E{exp[i(ϕ(x⃗)ϕ(y⃗))]}=exp(σϕ2)exp[Γϕ(r)].
M=exp(σϕ2)1(4πD2)22π0DrK0(r)exp[Γϕ(r)]dr1.
Γϕ(r)=2π0κJ0(κ·r)Φϕ(κ)F(κ)dκ,
F(κ)=1(2J1(κD2)κD2)2(4J2(κD2)κD2)2.
F(κ)=[1(2J1(κD2)κD2)2(4J2(κD2)κD2)2]·[1(2J1(κd2)κd2)2(4J2(κd2)κd2)2],
Φϕ(κ)=πk2L[1+kκ2Lsin(κ2Lk)]Φn(κ),
Φn(κ)=0.033Cn2κ11/3,
Γϕ(0)=σϕ2=0.134(Dr0)53,
M(D/r0)2

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