Abstract

A laser differential image-motion monitor (DIMM) system was designed and constructed as part of a turbulence characterization suite during the DARPA free-space optical experimental network experiment (FOENEX) program. The developed link measurement system measures the atmospheric coherence length (r0), atmospheric scintillation, and power in the bucket for the 1550 nm band. DIMM measurements are made with two separate apertures coupled to a single InGaAs camera. The angle of arrival (AoA) for the wavefront at each aperture can be calculated based on focal spot movements imaged by the camera. By utilizing a single camera for the simultaneous measurement of the focal spots, the correlation of the variance in the AoA allows a straightforward computation of r0 as in traditional DIMM systems. Standard measurements of scintillation and power in the bucket are made with the same apertures by redirecting a percentage of the incoming signals to InGaAs detectors integrated with logarithmic amplifiers for high sensitivity and high dynamic range. By leveraging two, small apertures, the instrument forms a small size and weight configuration for mounting to actively tracking laser communication terminals for characterizing link performance.

© 2013 Optical Society of America

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References

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  1. L. C. Andrews and R. L. Phillip, Laser Beam Propagation through Random Media (SPIE, 2005).
  2. J. C. Juarez, D. W. Young, R. A. Venkat, D. M. Brown, A. M. Brown, R. L. Oberc, J. E. Sluz, H. A. Pike, and L. B. Stotts, “Analysis of link performance for the FOENEX laser communications system,” Proc. SPIE 8380, 838007 (2012).
    [CrossRef]
  3. L. C. Andrews, R. L. Phillips, D. Wayne, P. Sauer, T. Leclerc, and R. Crabbs, “Creating a Cn2 profile as a function of altitude using scintillation measurements along a slant path,” Proc. SPIE 8238, 82380F (2012).
    [CrossRef]
  4. A. Tokovinin, “From differential image motion to seeing,” Publ. Astron. Soc. Pac. 114, 1156–1166 (2002).
    [CrossRef]
  5. D. Eaton, W. A. Peterson, J. R. Hines, J. J. Drexler, A. H. Walde, and D. B. Soules, “Comparison of two techniques for determining atmospheric seeing,” Proc. SPIE 926, 319–334 (1988).
    [CrossRef]
  6. M. S. Belenkii, D. W. Roberts, J. M. Stewart, G. G. Gimmestad, and W. R. Dagle, “Experimental validation of the differential image motion lidar concept,” Proc. SPIE 4377, 307–316 (2001).
    [CrossRef]
  7. V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).
  8. R. Fante, “Electromagnetic beam propagation in Turbulent Media,” Proc. IEEE 63, 1669–1692 (1975).
    [CrossRef]
  9. D. L. Knepp, “Multiple phase-screen calculation of the temporal behavior of stochastic waves,” Proc. IEEE 71, 722–737 (1983).
    [CrossRef]
  10. J. M. Martin and S. M. Flatté, “Intensity images and statistics from numerical simulation of wave propagation in 3-D random media,” Appl. Opt. 27, 2111–2126 (1988).
    [CrossRef]
  11. J. D. Schmidt, Numerical Simulation of Optical Wave Propagation with Examples in MATLAB (SPIE, 2010).
  12. A. M. Vorontsov, P. V. Paramonov, M. T. Valley, and M. A. Vorontsov, “Generation of infinitely long phase screens for modeling of optical wave propagation in atmospheric turbulence,” Waves Random Complex Media 18, 91–108 (2008).
    [CrossRef]
  13. B. M. Welsh, “A Fourier series based atmospheric phase screen generator for simulating anisoplanatic geometries and temporal evolution,” Proc. SPIE 3125, 327–338 (1997).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  17. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).
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    [CrossRef]
  19. L. C. Andrews, R. L. Phillips, D. Wayne, T. Leclerc, P. Sauer, R. Crabbs, and J. Kiriazes, “Near-ground vertical profile of refractive-index fluctuations,” Proc. SPIE 7324, 732402 (2009).
    [CrossRef]

2012 (2)

J. C. Juarez, D. W. Young, R. A. Venkat, D. M. Brown, A. M. Brown, R. L. Oberc, J. E. Sluz, H. A. Pike, and L. B. Stotts, “Analysis of link performance for the FOENEX laser communications system,” Proc. SPIE 8380, 838007 (2012).
[CrossRef]

L. C. Andrews, R. L. Phillips, D. Wayne, P. Sauer, T. Leclerc, and R. Crabbs, “Creating a Cn2 profile as a function of altitude using scintillation measurements along a slant path,” Proc. SPIE 8238, 82380F (2012).
[CrossRef]

2011 (1)

B. Felton and R. Alliss, “Improved climatological characterization of optical turbulence for free-space optical communications,” Proc. SPIE 8162, 816204 (2011).
[CrossRef]

2009 (1)

L. C. Andrews, R. L. Phillips, D. Wayne, T. Leclerc, P. Sauer, R. Crabbs, and J. Kiriazes, “Near-ground vertical profile of refractive-index fluctuations,” Proc. SPIE 7324, 732402 (2009).
[CrossRef]

2008 (1)

A. M. Vorontsov, P. V. Paramonov, M. T. Valley, and M. A. Vorontsov, “Generation of infinitely long phase screens for modeling of optical wave propagation in atmospheric turbulence,” Waves Random Complex Media 18, 91–108 (2008).
[CrossRef]

2002 (1)

A. Tokovinin, “From differential image motion to seeing,” Publ. Astron. Soc. Pac. 114, 1156–1166 (2002).
[CrossRef]

2001 (1)

M. S. Belenkii, D. W. Roberts, J. M. Stewart, G. G. Gimmestad, and W. R. Dagle, “Experimental validation of the differential image motion lidar concept,” Proc. SPIE 4377, 307–316 (2001).
[CrossRef]

1998 (1)

1997 (1)

B. M. Welsh, “A Fourier series based atmospheric phase screen generator for simulating anisoplanatic geometries and temporal evolution,” Proc. SPIE 3125, 327–338 (1997).
[CrossRef]

1995 (1)

1988 (2)

J. M. Martin and S. M. Flatté, “Intensity images and statistics from numerical simulation of wave propagation in 3-D random media,” Appl. Opt. 27, 2111–2126 (1988).
[CrossRef]

D. Eaton, W. A. Peterson, J. R. Hines, J. J. Drexler, A. H. Walde, and D. B. Soules, “Comparison of two techniques for determining atmospheric seeing,” Proc. SPIE 926, 319–334 (1988).
[CrossRef]

1983 (1)

D. L. Knepp, “Multiple phase-screen calculation of the temporal behavior of stochastic waves,” Proc. IEEE 71, 722–737 (1983).
[CrossRef]

1975 (1)

R. Fante, “Electromagnetic beam propagation in Turbulent Media,” Proc. IEEE 63, 1669–1692 (1975).
[CrossRef]

Alliss, R.

B. Felton and R. Alliss, “Improved climatological characterization of optical turbulence for free-space optical communications,” Proc. SPIE 8162, 816204 (2011).
[CrossRef]

Andrews, L. C.

L. C. Andrews, R. L. Phillips, D. Wayne, P. Sauer, T. Leclerc, and R. Crabbs, “Creating a Cn2 profile as a function of altitude using scintillation measurements along a slant path,” Proc. SPIE 8238, 82380F (2012).
[CrossRef]

L. C. Andrews, R. L. Phillips, D. Wayne, T. Leclerc, P. Sauer, R. Crabbs, and J. Kiriazes, “Near-ground vertical profile of refractive-index fluctuations,” Proc. SPIE 7324, 732402 (2009).
[CrossRef]

L. C. Andrews and R. L. Phillip, Laser Beam Propagation through Random Media (SPIE, 2005).

Belenkii, M. S.

M. S. Belenkii, D. W. Roberts, J. M. Stewart, G. G. Gimmestad, and W. R. Dagle, “Experimental validation of the differential image motion lidar concept,” Proc. SPIE 4377, 307–316 (2001).
[CrossRef]

Brown, A. M.

J. C. Juarez, D. W. Young, R. A. Venkat, D. M. Brown, A. M. Brown, R. L. Oberc, J. E. Sluz, H. A. Pike, and L. B. Stotts, “Analysis of link performance for the FOENEX laser communications system,” Proc. SPIE 8380, 838007 (2012).
[CrossRef]

Brown, D. M.

J. C. Juarez, D. W. Young, R. A. Venkat, D. M. Brown, A. M. Brown, R. L. Oberc, J. E. Sluz, H. A. Pike, and L. B. Stotts, “Analysis of link performance for the FOENEX laser communications system,” Proc. SPIE 8380, 838007 (2012).
[CrossRef]

Coles, W. A.

Crabbs, R.

L. C. Andrews, R. L. Phillips, D. Wayne, P. Sauer, T. Leclerc, and R. Crabbs, “Creating a Cn2 profile as a function of altitude using scintillation measurements along a slant path,” Proc. SPIE 8238, 82380F (2012).
[CrossRef]

L. C. Andrews, R. L. Phillips, D. Wayne, T. Leclerc, P. Sauer, R. Crabbs, and J. Kiriazes, “Near-ground vertical profile of refractive-index fluctuations,” Proc. SPIE 7324, 732402 (2009).
[CrossRef]

Dagle, W. R.

M. S. Belenkii, D. W. Roberts, J. M. Stewart, G. G. Gimmestad, and W. R. Dagle, “Experimental validation of the differential image motion lidar concept,” Proc. SPIE 4377, 307–316 (2001).
[CrossRef]

Drexler, J. J.

D. Eaton, W. A. Peterson, J. R. Hines, J. J. Drexler, A. H. Walde, and D. B. Soules, “Comparison of two techniques for determining atmospheric seeing,” Proc. SPIE 926, 319–334 (1988).
[CrossRef]

Eaton, D.

D. Eaton, W. A. Peterson, J. R. Hines, J. J. Drexler, A. H. Walde, and D. B. Soules, “Comparison of two techniques for determining atmospheric seeing,” Proc. SPIE 926, 319–334 (1988).
[CrossRef]

Fante, R.

R. Fante, “Electromagnetic beam propagation in Turbulent Media,” Proc. IEEE 63, 1669–1692 (1975).
[CrossRef]

Felton, B.

B. Felton and R. Alliss, “Improved climatological characterization of optical turbulence for free-space optical communications,” Proc. SPIE 8162, 816204 (2011).
[CrossRef]

Filice, J. P.

Flatté, S. M.

Fraser, D.

Frehlich, R. G.

Gimmestad, G. G.

M. S. Belenkii, D. W. Roberts, J. M. Stewart, G. G. Gimmestad, and W. R. Dagle, “Experimental validation of the differential image motion lidar concept,” Proc. SPIE 4377, 307–316 (2001).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

Hines, J. R.

D. Eaton, W. A. Peterson, J. R. Hines, J. J. Drexler, A. H. Walde, and D. B. Soules, “Comparison of two techniques for determining atmospheric seeing,” Proc. SPIE 926, 319–334 (1988).
[CrossRef]

Juarez, J. C.

J. C. Juarez, D. W. Young, R. A. Venkat, D. M. Brown, A. M. Brown, R. L. Oberc, J. E. Sluz, H. A. Pike, and L. B. Stotts, “Analysis of link performance for the FOENEX laser communications system,” Proc. SPIE 8380, 838007 (2012).
[CrossRef]

Kiriazes, J.

L. C. Andrews, R. L. Phillips, D. Wayne, T. Leclerc, P. Sauer, R. Crabbs, and J. Kiriazes, “Near-ground vertical profile of refractive-index fluctuations,” Proc. SPIE 7324, 732402 (2009).
[CrossRef]

Knepp, D. L.

D. L. Knepp, “Multiple phase-screen calculation of the temporal behavior of stochastic waves,” Proc. IEEE 71, 722–737 (1983).
[CrossRef]

Lambert, A. J.

Leclerc, T.

L. C. Andrews, R. L. Phillips, D. Wayne, P. Sauer, T. Leclerc, and R. Crabbs, “Creating a Cn2 profile as a function of altitude using scintillation measurements along a slant path,” Proc. SPIE 8238, 82380F (2012).
[CrossRef]

L. C. Andrews, R. L. Phillips, D. Wayne, T. Leclerc, P. Sauer, R. Crabbs, and J. Kiriazes, “Near-ground vertical profile of refractive-index fluctuations,” Proc. SPIE 7324, 732402 (2009).
[CrossRef]

Martin, J. M.

Oberc, R. L.

J. C. Juarez, D. W. Young, R. A. Venkat, D. M. Brown, A. M. Brown, R. L. Oberc, J. E. Sluz, H. A. Pike, and L. B. Stotts, “Analysis of link performance for the FOENEX laser communications system,” Proc. SPIE 8380, 838007 (2012).
[CrossRef]

Paramonov, P. V.

A. M. Vorontsov, P. V. Paramonov, M. T. Valley, and M. A. Vorontsov, “Generation of infinitely long phase screens for modeling of optical wave propagation in atmospheric turbulence,” Waves Random Complex Media 18, 91–108 (2008).
[CrossRef]

Peterson, W. A.

D. Eaton, W. A. Peterson, J. R. Hines, J. J. Drexler, A. H. Walde, and D. B. Soules, “Comparison of two techniques for determining atmospheric seeing,” Proc. SPIE 926, 319–334 (1988).
[CrossRef]

Phillip, R. L.

L. C. Andrews and R. L. Phillip, Laser Beam Propagation through Random Media (SPIE, 2005).

Phillips, R. L.

L. C. Andrews, R. L. Phillips, D. Wayne, P. Sauer, T. Leclerc, and R. Crabbs, “Creating a Cn2 profile as a function of altitude using scintillation measurements along a slant path,” Proc. SPIE 8238, 82380F (2012).
[CrossRef]

L. C. Andrews, R. L. Phillips, D. Wayne, T. Leclerc, P. Sauer, R. Crabbs, and J. Kiriazes, “Near-ground vertical profile of refractive-index fluctuations,” Proc. SPIE 7324, 732402 (2009).
[CrossRef]

Pike, H. A.

J. C. Juarez, D. W. Young, R. A. Venkat, D. M. Brown, A. M. Brown, R. L. Oberc, J. E. Sluz, H. A. Pike, and L. B. Stotts, “Analysis of link performance for the FOENEX laser communications system,” Proc. SPIE 8380, 838007 (2012).
[CrossRef]

Roberts, D. W.

M. S. Belenkii, D. W. Roberts, J. M. Stewart, G. G. Gimmestad, and W. R. Dagle, “Experimental validation of the differential image motion lidar concept,” Proc. SPIE 4377, 307–316 (2001).
[CrossRef]

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007).

Sauer, P.

L. C. Andrews, R. L. Phillips, D. Wayne, P. Sauer, T. Leclerc, and R. Crabbs, “Creating a Cn2 profile as a function of altitude using scintillation measurements along a slant path,” Proc. SPIE 8238, 82380F (2012).
[CrossRef]

L. C. Andrews, R. L. Phillips, D. Wayne, T. Leclerc, P. Sauer, R. Crabbs, and J. Kiriazes, “Near-ground vertical profile of refractive-index fluctuations,” Proc. SPIE 7324, 732402 (2009).
[CrossRef]

Schmidt, J. D.

J. D. Schmidt, Numerical Simulation of Optical Wave Propagation with Examples in MATLAB (SPIE, 2010).

Sluz, J. E.

J. C. Juarez, D. W. Young, R. A. Venkat, D. M. Brown, A. M. Brown, R. L. Oberc, J. E. Sluz, H. A. Pike, and L. B. Stotts, “Analysis of link performance for the FOENEX laser communications system,” Proc. SPIE 8380, 838007 (2012).
[CrossRef]

Soules, D. B.

D. Eaton, W. A. Peterson, J. R. Hines, J. J. Drexler, A. H. Walde, and D. B. Soules, “Comparison of two techniques for determining atmospheric seeing,” Proc. SPIE 926, 319–334 (1988).
[CrossRef]

Stewart, J. M.

M. S. Belenkii, D. W. Roberts, J. M. Stewart, G. G. Gimmestad, and W. R. Dagle, “Experimental validation of the differential image motion lidar concept,” Proc. SPIE 4377, 307–316 (2001).
[CrossRef]

Stotts, L. B.

J. C. Juarez, D. W. Young, R. A. Venkat, D. M. Brown, A. M. Brown, R. L. Oberc, J. E. Sluz, H. A. Pike, and L. B. Stotts, “Analysis of link performance for the FOENEX laser communications system,” Proc. SPIE 8380, 838007 (2012).
[CrossRef]

Tatarskii, V. I.

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

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007).

Tokovinin, A.

A. Tokovinin, “From differential image motion to seeing,” Publ. Astron. Soc. Pac. 114, 1156–1166 (2002).
[CrossRef]

Valley, M. T.

A. M. Vorontsov, P. V. Paramonov, M. T. Valley, and M. A. Vorontsov, “Generation of infinitely long phase screens for modeling of optical wave propagation in atmospheric turbulence,” Waves Random Complex Media 18, 91–108 (2008).
[CrossRef]

Venkat, R. A.

J. C. Juarez, D. W. Young, R. A. Venkat, D. M. Brown, A. M. Brown, R. L. Oberc, J. E. Sluz, H. A. Pike, and L. B. Stotts, “Analysis of link performance for the FOENEX laser communications system,” Proc. SPIE 8380, 838007 (2012).
[CrossRef]

Vorontsov, A. M.

A. M. Vorontsov, P. V. Paramonov, M. T. Valley, and M. A. Vorontsov, “Generation of infinitely long phase screens for modeling of optical wave propagation in atmospheric turbulence,” Waves Random Complex Media 18, 91–108 (2008).
[CrossRef]

Vorontsov, M. A.

A. M. Vorontsov, P. V. Paramonov, M. T. Valley, and M. A. Vorontsov, “Generation of infinitely long phase screens for modeling of optical wave propagation in atmospheric turbulence,” Waves Random Complex Media 18, 91–108 (2008).
[CrossRef]

Walde, A. H.

D. Eaton, W. A. Peterson, J. R. Hines, J. J. Drexler, A. H. Walde, and D. B. Soules, “Comparison of two techniques for determining atmospheric seeing,” Proc. SPIE 926, 319–334 (1988).
[CrossRef]

Wayne, D.

L. C. Andrews, R. L. Phillips, D. Wayne, P. Sauer, T. Leclerc, and R. Crabbs, “Creating a Cn2 profile as a function of altitude using scintillation measurements along a slant path,” Proc. SPIE 8238, 82380F (2012).
[CrossRef]

L. C. Andrews, R. L. Phillips, D. Wayne, T. Leclerc, P. Sauer, R. Crabbs, and J. Kiriazes, “Near-ground vertical profile of refractive-index fluctuations,” Proc. SPIE 7324, 732402 (2009).
[CrossRef]

Welsh, B. M.

B. M. Welsh, “A Fourier series based atmospheric phase screen generator for simulating anisoplanatic geometries and temporal evolution,” Proc. SPIE 3125, 327–338 (1997).
[CrossRef]

Yadlowsky, M.

Young, D. W.

J. C. Juarez, D. W. Young, R. A. Venkat, D. M. Brown, A. M. Brown, R. L. Oberc, J. E. Sluz, H. A. Pike, and L. B. Stotts, “Analysis of link performance for the FOENEX laser communications system,” Proc. SPIE 8380, 838007 (2012).
[CrossRef]

Appl. Opt. (3)

Proc. IEEE (2)

R. Fante, “Electromagnetic beam propagation in Turbulent Media,” Proc. IEEE 63, 1669–1692 (1975).
[CrossRef]

D. L. Knepp, “Multiple phase-screen calculation of the temporal behavior of stochastic waves,” Proc. IEEE 71, 722–737 (1983).
[CrossRef]

Proc. SPIE (7)

J. C. Juarez, D. W. Young, R. A. Venkat, D. M. Brown, A. M. Brown, R. L. Oberc, J. E. Sluz, H. A. Pike, and L. B. Stotts, “Analysis of link performance for the FOENEX laser communications system,” Proc. SPIE 8380, 838007 (2012).
[CrossRef]

L. C. Andrews, R. L. Phillips, D. Wayne, P. Sauer, T. Leclerc, and R. Crabbs, “Creating a Cn2 profile as a function of altitude using scintillation measurements along a slant path,” Proc. SPIE 8238, 82380F (2012).
[CrossRef]

D. Eaton, W. A. Peterson, J. R. Hines, J. J. Drexler, A. H. Walde, and D. B. Soules, “Comparison of two techniques for determining atmospheric seeing,” Proc. SPIE 926, 319–334 (1988).
[CrossRef]

M. S. Belenkii, D. W. Roberts, J. M. Stewart, G. G. Gimmestad, and W. R. Dagle, “Experimental validation of the differential image motion lidar concept,” Proc. SPIE 4377, 307–316 (2001).
[CrossRef]

B. M. Welsh, “A Fourier series based atmospheric phase screen generator for simulating anisoplanatic geometries and temporal evolution,” Proc. SPIE 3125, 327–338 (1997).
[CrossRef]

B. Felton and R. Alliss, “Improved climatological characterization of optical turbulence for free-space optical communications,” Proc. SPIE 8162, 816204 (2011).
[CrossRef]

L. C. Andrews, R. L. Phillips, D. Wayne, T. Leclerc, P. Sauer, R. Crabbs, and J. Kiriazes, “Near-ground vertical profile of refractive-index fluctuations,” Proc. SPIE 7324, 732402 (2009).
[CrossRef]

Publ. Astron. Soc. Pac. (1)

A. Tokovinin, “From differential image motion to seeing,” Publ. Astron. Soc. Pac. 114, 1156–1166 (2002).
[CrossRef]

Waves Random Complex Media (1)

A. M. Vorontsov, P. V. Paramonov, M. T. Valley, and M. A. Vorontsov, “Generation of infinitely long phase screens for modeling of optical wave propagation in atmospheric turbulence,” Waves Random Complex Media 18, 91–108 (2008).
[CrossRef]

Other (5)

L. C. Andrews and R. L. Phillip, Laser Beam Propagation through Random Media (SPIE, 2005).

J. D. Schmidt, Numerical Simulation of Optical Wave Propagation with Examples in MATLAB (SPIE, 2010).

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

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

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Figures (7)

Fig. 1.
Fig. 1.

Split-step propagation through atmospheric turbulence.

Fig. 2.
Fig. 2.

Propagation internal to the DIMM receiver and calculation of atmospheric coherence length using simulated DIMM data.

Fig. 3.
Fig. 3.

Wave propagation simulation of proposed DIMM hardware using 500 independent realizations of turbulence. The boxed hardware configuration was constructed for testing.

Fig. 4.
Fig. 4.

Optical layout (left) and mechanical design (right) of the DIMM sensor.

Fig. 5.
Fig. 5.

DIMM sensor deployed in the field for turbulence characterization during the 10 km link testing (left) [2]. System diagram for the 10 km link testing for the FOENEX program (right).

Fig. 6.
Fig. 6.

Example DIMM data collected during a recent field-testing exercise. Two individual frames captured at different times (top to bottom) during low turbulence (left) and high turbulence (right).

Fig. 7.
Fig. 7.

Processed DIMM data compared to other measurements and model results [2].

Equations (9)

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

ϕ(x,y)=Ψ(fx,fy)ei2π(fxx+fyy)dfxdfy,
ϕ(x,y)=n=m=cn,mexp[i2π(fxnx+fymy)],
|cn,m|2=Φϕ(fxn,fym)ΔfxnΔfym.
U(Ar2)=1iλBeiπβr22[U(r1)eiπαr12],
r0=38[f2λ2(0.697d1/30.484D1/3)σXY2]3/5[m].
Cn2=(0.423k2r05/3L)1[m2/3].
r⃗lft=r⃗lft,home+r⃗lft,vib+r⃗lft,turbr⃗rgt=r⃗rgt,home+r⃗rgt,vib+r⃗rgt,turb,
r⃗turb=r⃗lft,turbr⃗rgt,turb=r⃗lftrlft,homer⃗lft,vib(r⃗rgtr⃗rgt,homer⃗rgt,vib),
σXY2=r¯turb2r¯turb2.

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