Abstract

The split-step Fourier-transform algorithm for numerical simulation of wave propagation in a turbulent atmosphere is refined to correctly include the effects of large-scale phase fluctuations that are important for imaging problems and many beam-wave problems such as focused laser beams and beam spreading. The results of the improved algorithm are similar to the results of the traditional algorithm for the performance of coherent Doppler lidar and for plane-wave intensity statistics because the effects of large-scale turbulence are less important. The series solution for coherent Doppler lidar performance converges slowly to the results from simulation.

© 2000 Optical Society of America

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References

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  1. V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Keter Press, Jerusalem, 1971).
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    [CrossRef]
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    [CrossRef]
  4. D. L. Knepp, “Multiple phase-screen calculation of the temporal behavior of stochastic waves,” Proc. IEEE 71, 722–737 (1983).
    [CrossRef]
  5. C. Macaskill, T. E. Ewart, “Computer simulation of two-dimensional random wave propagation,” IMA J. Appl. Math. 33, 1–15 (1984).
    [CrossRef]
  6. J. Martin, S. Flatté, “Intensity images and statistics from numerical simulation of wave propagation in 3-D random media,” Appl. Opt. 27, 2111–2126 (1988).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  15. A. Belmonte, B. J. Rye, W. A. Brewer, R. M. Hardesty, “Coherent lidar returns in turbulent atmosphere from simulation of beam propagation,” in Coherent Laser Radar Technology and Applications Conference (Universities Space Research Association, 10227 Wincopin Circle, Columbia, MD 21044-3498, 1999).
  16. C. L. Rino, J. Owen, “Numerical simulations of intensity scintillation using the power law phase screen model,” Radio Sci. 19, 891–908 (1984).
    [CrossRef]
  17. W. A. Coles, J. P. Filice, “Dynamic spectra of interplanetary scintillations,” Nature (London) 312, 251–254 (1985).
    [CrossRef]
  18. Wm. A. Coles, J. P. Filice, R. G. Frehlich, M. Yadlowsky, “Simulation of wave propagation in three-dimensional random media,” Appl. Opt. 34, 2089–2101 (1995).
    [CrossRef] [PubMed]
  19. R. J. Hill, “Models of the scalar spectrum for turbulent advection,” J. Fluid Mech. 88, 541–562 (1978).
    [CrossRef]
  20. R. Frehlich, “Laser scintillation measurements of the temperature spectrum in the atmospheric surface layer,” J. Atmos. Sci. 49, 1494–1509 (1992).
    [CrossRef]
  21. N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
    [CrossRef]
  22. R. G. Lane, A. Glindemann, J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
    [CrossRef]
  23. A. Ishimaru, “Fluctuations of a beam wave propagating through a locally homogeneous medium,” Radio Sci. 4, 295–305 (1969).
    [CrossRef]
  24. R. G. Frehlich, M. J. Kavaya, “Coherent laser radar performance for general atmospheric refractive turbulence,” Appl. Opt. 30, 5325–5352 (1991).
    [CrossRef] [PubMed]
  25. F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1981), Vol. 19, pp. 280–370.

1997

1996

1995

1994

1993

1992

R. Frehlich, “Laser scintillation measurements of the temperature spectrum in the atmospheric surface layer,” J. Atmos. Sci. 49, 1494–1509 (1992).
[CrossRef]

R. G. Lane, A. Glindemann, J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
[CrossRef]

1991

1990

1988

1985

W. A. Coles, J. P. Filice, “Dynamic spectra of interplanetary scintillations,” Nature (London) 312, 251–254 (1985).
[CrossRef]

1984

C. Macaskill, T. E. Ewart, “Computer simulation of two-dimensional random wave propagation,” IMA J. Appl. Math. 33, 1–15 (1984).
[CrossRef]

C. L. Rino, J. Owen, “Numerical simulations of intensity scintillation using the power law phase screen model,” Radio Sci. 19, 891–908 (1984).
[CrossRef]

1983

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

1978

R. J. Hill, “Models of the scalar spectrum for turbulent advection,” J. Fluid Mech. 88, 541–562 (1978).
[CrossRef]

1976

J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

1975

S. M. Flatté, F. D. Tappert, “Calculation of the effect of internal waves on oceanic sound transmission,” J. Acoust. Soc. Am. 58, 1151–1159 (1975).
[CrossRef]

1969

A. Ishimaru, “Fluctuations of a beam wave propagating through a locally homogeneous medium,” Radio Sci. 4, 295–305 (1969).
[CrossRef]

Belmonte, A.

A. Belmonte, B. J. Rye, W. A. Brewer, R. M. Hardesty, “Coherent lidar returns in turbulent atmosphere from simulation of beam propagation,” in Coherent Laser Radar Technology and Applications Conference (Universities Space Research Association, 10227 Wincopin Circle, Columbia, MD 21044-3498, 1999).

Bracher, C.

Brewer, W. A.

A. Belmonte, B. J. Rye, W. A. Brewer, R. M. Hardesty, “Coherent lidar returns in turbulent atmosphere from simulation of beam propagation,” in Coherent Laser Radar Technology and Applications Conference (Universities Space Research Association, 10227 Wincopin Circle, Columbia, MD 21044-3498, 1999).

Coles, W. A.

W. A. Coles, J. P. Filice, “Dynamic spectra of interplanetary scintillations,” Nature (London) 312, 251–254 (1985).
[CrossRef]

Coles, Wm. A.

Dainty, J. C.

R. G. Lane, A. Glindemann, J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
[CrossRef]

Ewart, T. E.

C. Macaskill, T. E. Ewart, “Computer simulation of two-dimensional random wave propagation,” IMA J. Appl. Math. 33, 1–15 (1984).
[CrossRef]

Feit, M. D.

J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

Filice, J. P.

Flatté, S.

Flatté, S. M.

Fleck, J. A.

J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

Frehlich, R.

R. Frehlich, “Laser scintillation measurements of the temperature spectrum in the atmospheric surface layer,” J. Atmos. Sci. 49, 1494–1509 (1992).
[CrossRef]

Frehlich, R. G.

Glindemann, A.

R. G. Lane, A. Glindemann, J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
[CrossRef]

Gudimetla, V. S.

D. G. Youmans, V. S. Gudimetla, “Round-trip turbulence scintillation effects on laser radar: Monte Carlo simulation results for unresolved targets,” in Laser Radar Technology and Applications II, G. W. Kamerman, ed. Proc. SPIE3065, 71–83 (1997).
[CrossRef]

Hardesty, R. M.

A. Belmonte, B. J. Rye, W. A. Brewer, R. M. Hardesty, “Coherent lidar returns in turbulent atmosphere from simulation of beam propagation,” in Coherent Laser Radar Technology and Applications Conference (Universities Space Research Association, 10227 Wincopin Circle, Columbia, MD 21044-3498, 1999).

Hill, R. J.

Ishimaru, A.

A. Ishimaru, “Fluctuations of a beam wave propagating through a locally homogeneous medium,” Radio Sci. 4, 295–305 (1969).
[CrossRef]

Kavaya, M. J.

Knepp, D. L.

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

Lane, R. G.

R. G. Lane, A. Glindemann, J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
[CrossRef]

Macaskill, C.

C. Macaskill, T. E. Ewart, “Computer simulation of two-dimensional random wave propagation,” IMA J. Appl. Math. 33, 1–15 (1984).
[CrossRef]

Martin, J.

Martin, J. M.

Morris, J. R.

J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

Owen, J.

C. L. Rino, J. Owen, “Numerical simulations of intensity scintillation using the power law phase screen model,” Radio Sci. 19, 891–908 (1984).
[CrossRef]

Rino, C. L.

C. L. Rino, J. Owen, “Numerical simulations of intensity scintillation using the power law phase screen model,” Radio Sci. 19, 891–908 (1984).
[CrossRef]

Roddier, F.

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1981), Vol. 19, pp. 280–370.

Roddier, N.

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

Rye, B. J.

A. Belmonte, B. J. Rye, W. A. Brewer, R. M. Hardesty, “Coherent lidar returns in turbulent atmosphere from simulation of beam propagation,” in Coherent Laser Radar Technology and Applications Conference (Universities Space Research Association, 10227 Wincopin Circle, Columbia, MD 21044-3498, 1999).

Tappert, F. D.

S. M. Flatté, F. D. Tappert, “Calculation of the effect of internal waves on oceanic sound transmission,” J. Acoust. Soc. Am. 58, 1151–1159 (1975).
[CrossRef]

Tatarskii, V. I.

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Keter Press, Jerusalem, 1971).

Wang, G.

Wang, G-Yu

Yadlowsky, M.

Youmans, D. G.

D. G. Youmans, V. S. Gudimetla, “Round-trip turbulence scintillation effects on laser radar: Monte Carlo simulation results for unresolved targets,” in Laser Radar Technology and Applications II, G. W. Kamerman, ed. Proc. SPIE3065, 71–83 (1997).
[CrossRef]

Appl. Opt.

Appl. Phys.

J. A. Fleck, J. R. Morris, M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. 10, 129–160 (1976).
[CrossRef]

IMA J. Appl. Math.

C. Macaskill, T. E. Ewart, “Computer simulation of two-dimensional random wave propagation,” IMA J. Appl. Math. 33, 1–15 (1984).
[CrossRef]

J. Acoust. Soc. Am.

S. M. Flatté, F. D. Tappert, “Calculation of the effect of internal waves on oceanic sound transmission,” J. Acoust. Soc. Am. 58, 1151–1159 (1975).
[CrossRef]

J. Atmos. Sci.

R. Frehlich, “Laser scintillation measurements of the temperature spectrum in the atmospheric surface layer,” J. Atmos. Sci. 49, 1494–1509 (1992).
[CrossRef]

J. Fluid Mech.

R. J. Hill, “Models of the scalar spectrum for turbulent advection,” J. Fluid Mech. 88, 541–562 (1978).
[CrossRef]

J. Opt. Soc. Am. A

Nature (London)

W. A. Coles, J. P. Filice, “Dynamic spectra of interplanetary scintillations,” Nature (London) 312, 251–254 (1985).
[CrossRef]

Opt. Eng.

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

Proc. IEEE

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

Radio Sci.

C. L. Rino, J. Owen, “Numerical simulations of intensity scintillation using the power law phase screen model,” Radio Sci. 19, 891–908 (1984).
[CrossRef]

A. Ishimaru, “Fluctuations of a beam wave propagating through a locally homogeneous medium,” Radio Sci. 4, 295–305 (1969).
[CrossRef]

Waves Random Media

R. G. Lane, A. Glindemann, J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
[CrossRef]

Other

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1981), Vol. 19, pp. 280–370.

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (Keter Press, Jerusalem, 1971).

J. Martin, “Simulation of wave propagation in random: theory and applications,” in Wave Propagation in Random Media (Scintillation), V. I. Tatarskii, A. Ishimaru, V. U. Zavorotny, eds. (SPIE Press, Bellingham, Wash., 1993).

D. G. Youmans, V. S. Gudimetla, “Round-trip turbulence scintillation effects on laser radar: Monte Carlo simulation results for unresolved targets,” in Laser Radar Technology and Applications II, G. W. Kamerman, ed. Proc. SPIE3065, 71–83 (1997).
[CrossRef]

A. Belmonte, B. J. Rye, W. A. Brewer, R. M. Hardesty, “Coherent lidar returns in turbulent atmosphere from simulation of beam propagation,” in Coherent Laser Radar Technology and Applications Conference (Universities Space Research Association, 10227 Wincopin Circle, Columbia, MD 21044-3498, 1999).

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Figures (6)

Fig. 1
Fig. 1

Phase structure functions D θ(kΔ) versus lag k from FFT simulation (dotted curve), improved FFT–SH (subharmonic) simulation with N P = 3 orders of subharmonics (dashed curve), and input theoretical model (solid curve) Eq. (5) for a N = 512 × 512 simulation and inner scale l 0 = 5Δ.

Fig. 2
Fig. 2

Average intensity profile 〈I(r)〉 of a focused Gaussian beam transmitted through a circular aperture from a N = 256 × 256 FFT simulation (open circles), N = 512 × 512 FFT simulation (open squares), N = 256 × 256 FFT–SH simulation (filled circles), and the theoretical value (solid curve) from Eqs. (10) and (11).

Fig. 3
Fig. 3

Normalized variance profile m 2(r) Eq. (12) for the same parameters as Fig. 2 and various square simulation sizes N.

Fig. 4
Fig. 4

Coherent Doppler lidar performance η S Eq. (13) versus turbulence level C n 2 for the same parameters as Fig. 2. The leading-order term η S0 Eq. (14) of the theoretical expansion for η S in weak turbulence (low C n 2) is shown as a dotted curve and the leading-order term η S1 = 2η S0 for strong turbulence (high C n 2) is shown as a dashed curve. The solid line indicates SD[η S ] ∝ C n 2 as required by the Born or the Rytov theory.

Fig. 5
Fig. 5

The MCF Eq. (15) for a plane-wave simulation compared with theory (solid curve) Eq. (16) for N = 512, Δ = 2.0 mm, λ = 1.0 µm, l 0 = 5.0 mm, C n 2 = 1.0 × 10-15 m-2/3, and a propagation distance z = 500 m.

Fig. 6
Fig. 6

Normalized variance m 2 versus turbulence level C n 2 for a plane-wave geometry with λ = 1 µm, z = 500 m, l 0 = 5 mm, and Δ = 0.5 mm.

Equations (17)

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θjΔx, lΔy=n=0Nxm=0Nyan, m+ibn, m×exp2πijn/Nx+lm/Ny,
a2n, m=b2n, m=ΔqxΔqyΦθnΔqx, mΔqy, z,
Φθqx, qy, z=2πk2ΔzΦnqx, qy, qz=0, z,
Φnq, z=ACn2zq-11/3fql0z,
Dθs, z=θr-θr+s2=2 -1-coss·q×Φθq, zdq=Δzds, z,
ds, z=4πk2-1-coss·q×Φnqx, qy, qz=0, zdq
θSHjΔx, lΔy=p=1NPn=-11m=-11an, m, p+ibn, m, p×exp2πijn/3pNx+2πilm/3pNy,
a2n, m, p=b2n, m, p=ΔqxpΔqypΦθnΔqxp, mΔqyp
a2n, m, p=b2n, m, p=n-1/2Δqxpn+1/2Δqxpm-1/2Δqypm+1/2Δqyp Φθqx, qydqydqx,
Ir, z=k22πz2- Osexp-ikr·s/z×exp -120zds1-u/z, ududs,
Os, z=- Er+s/2, 0E*r-s/2, 0× expikr·s/zdr.
m2r=Ir-Ir2Ir2,
ηSz=λ2z2AR- IT2p, zdp,
ηS0z=1AR- |Os, z|2×exp-0zds1-u/z, ududs,
MCFs, z=Er, zE*r+s, z.
MCFs, z=Iexp-Dθs/2
m2r=Ir-Ir2Ir2=Ir-I2I2

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