Abstract

The relationship between stellar scintillation in the strong-focusing regime and the atmosphere’s vertical turbulence profile is investigated with numerical simulation. For two distinct atmospheric profiles, the irradiance variance at a point on a telescope aperture is evaluated as a function of the weighted path-integrated turbulence (i.e., Rytov variance). Additionally, we compute the aperture-averaged irradiance variance and the log-amplitude correlation across the aperture as functions of the Rytov variance. For one atmospheric profile, scintillation is dominated by turbulence in the tropopause; for the other, scintillation arises from turbulence in both the tropopause and the lower troposphere. The numerical results indicate that (1) stellar scintillation depends on the actual profile of atmospheric turbulence and not just on its weighted integral and (2) in the strong-focusing regime the irradiance variance is determined primarily by an optical wave’s coherence length as it passes through the tropopause.

© 2001 Optical Society of America

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References

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  1. R. R. Beland, “Propagation through atmospheric optical turbulence,” in Atmospheric Propagation of Radiation, F. G. Smith, ed. (SPIE Press, Bellingham, Wash., 1993), Chap. 2.
  2. V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961), Chap. 8.
  3. M. E. Gracheva, A. S. Gurvich, S. S. Kashkarov, Vl. V. Pokasov, “Similarity relations and their experimental verification for strong intensity fluctuations of laser radiation,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, Berlin, 1978), Chap. 4.
  4. A. Consortini, F. Cochetti, J. H. Churnside, R. J. Hill, “Inner-scale effect on irradiance variance measured for weak-to-strong atmospheric scintillation,” J. Opt. Soc. Am. A 10, 2354–2362 (1993).
    [CrossRef]
  5. S. M. Flatté, G.-Y. Wang, J. Martin, “Irradiance variance of optical waves through atmospheric turbulence by numerical simulation and comparison with experiment,” J. Opt. Soc. Am. A 10, 2363–2370 (1993).
    [CrossRef]
  6. S. M. Flatté, C. Bracher, 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);R. Hill, R. G. Frehlich, “Probability distribution of irradiance for the onset of strong scintillation,” J. Opt. Soc. Am. A 14, 1530–1540 (1997).
    [CrossRef]
  7. D. Dravins, L. Lindegren, E. Mezey, A. T. Young, “Atmospheric intensity scintillation of stars. I. Statistical distributions and temporal properties,” Publ. Astron. Soc. Pac. 109, 173–207 (1997).
    [CrossRef]
  8. J. Bufton, S. H. Genatt, “Simultaneous observations of atmospheric turbulence effects on stellar irradiance and phase,” Astron. J. 76, 378–386 (1971); see also G. Parry, J. G. Walker, R. J. Scaddan, “On the statistics of stellar speckle patterns and pupil plane scintillation,” Opt. Acta 26, 563–574 (1979).
    [CrossRef]
  9. In the following, we refer to the lower 9 km of atmosphere as the troposphere, the 9–12-km region as the tropopause, and the 12–20-km region as the lower stratosphere.
  10. H. A. Whale, “Diffraction of a plane wave by a random phase screen,” J. Atmos. Terr. Phys. 35, 263–274 (1973);R. Buckley, “Diffraction by a random phase-changing screen: a numerical experiment,” J. Atmos. Terr. Phys. 37, 1431–1446 (1975);W. P. Brown, “Computer simulation of adaptive optical systems,” Hughes Research Laboratories Report, contract N60921–74–C–0249, 1975; B. L. McGlamery, “Computer simulation studies of compensation of turbulence degraded images,” in Image Processing, J. C. Urbach, ed., Proc. SPIE74, 225–233 (1976)
    [CrossRef]
  11. J. M. Martin, S. M. Flatté, “Intensity images and statistics from numerical simulation of wave propagation in 3-D random media,” Appl. Opt. 27, 2111–2126 (1988);W. 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]
  12. J. W. Goodman, Introduction to Fourier Optics (McGraw–Hill, San Francisco, Calif., 1968).
  13. For completeness, we note that the pixel size of ∼0.3 cm–0.4 cm introduces an inner scale of turbulence into the simulations. This is an unavoidable consequence of all finite element numerical simulations.
  14. In principle, a telescope of any diameter may be simulated. However, for larger-aperture telescopes to be considered, the phase-screen length must also become larger, which in turn requires a larger number of pixels to span the screen’s length. (As discussed in Appendix A, the pixel size is bounded by considerations related to the strength of turbulence and the interscreen spacing.) The computational requirements for dealing with phase screens that have large numbers of pixels can quickly become prohibitive.
  15. William L. Hays, Statistics (Holt, Rinehart & Winston, New York, 1988), Chap. 5.
  16. J. W. Strohbehn, “Modern theories in the propagation of optical waves in a turbulent medium,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn ed. (Springer–Verlag, Berlin, 1978), Chap. 3.
  17. H. T. Yura, “Physical model for strong optical-amplitude fluctuations in a turbulent medium,” J. Opt. Soc. Am. 64, 59 (1974).
    [CrossRef]
  18. S. F. Clifford, G. R. Ochs, R. S. Lawrence, “Saturation of optical scintillation by strong turbulence,” J. Opt. Soc. Am. 64, 148–154 (1974).
    [CrossRef]
  19. R. E. Hufnagel, N. R. Stanley, “Modulation transfer function associated with image transmission through turbulent media,” J. Opt. Soc. Am. 54, 52–61 (1964).
    [CrossRef]
  20. S. E. Troxel, R. M. Welsh, M. C. Roggemann, “Off-axis optical transfer function calculations in an adaptive-optics system by means of a diffraction calculation for weak index fluctuations,” J. Opt. Soc. Am. A 11, 2100–2111 (1994).
    [CrossRef]
  21. C-E. Fröberg, Introduction to Numerical Analysis (Addison–Wesley, Reading, Mass., 1965), Chap. 10.
  22. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, (Dover, New York, 1972), Chap. 25.
  23. The author received the text of D. Fried’s phase-screen generation algorithm from D. L. Fried, 14671 Tumbleweed Lane, Monterey County, Calif. 95076 (private communication).
  24. M. J. Levin, “Generation of a sampled Gaussian time series having a specified correlation function,” IRE Trans. Inf. Theory IT-6, 545–548 (1960);J. C. Camparo, P. Lambropoulos, “Monte Carlo simulations of field fluctuations in strongly driven resonant transitions,” Phys. Rev. A 47, 480–494 (1993).
    [CrossRef] [PubMed]
  25. D. L. Knepp, “Multiple phase-screen calculation of the temporal behavior of stochastic waves,” Proc. IEEE 71, 722–737 (1983).
    [CrossRef]

1997 (1)

D. Dravins, L. Lindegren, E. Mezey, A. T. Young, “Atmospheric intensity scintillation of stars. I. Statistical distributions and temporal properties,” Publ. Astron. Soc. Pac. 109, 173–207 (1997).
[CrossRef]

1994 (2)

1993 (2)

1988 (1)

1983 (1)

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

1974 (2)

1973 (1)

H. A. Whale, “Diffraction of a plane wave by a random phase screen,” J. Atmos. Terr. Phys. 35, 263–274 (1973);R. Buckley, “Diffraction by a random phase-changing screen: a numerical experiment,” J. Atmos. Terr. Phys. 37, 1431–1446 (1975);W. P. Brown, “Computer simulation of adaptive optical systems,” Hughes Research Laboratories Report, contract N60921–74–C–0249, 1975; B. L. McGlamery, “Computer simulation studies of compensation of turbulence degraded images,” in Image Processing, J. C. Urbach, ed., Proc. SPIE74, 225–233 (1976)
[CrossRef]

1971 (1)

J. Bufton, S. H. Genatt, “Simultaneous observations of atmospheric turbulence effects on stellar irradiance and phase,” Astron. J. 76, 378–386 (1971); see also G. Parry, J. G. Walker, R. J. Scaddan, “On the statistics of stellar speckle patterns and pupil plane scintillation,” Opt. Acta 26, 563–574 (1979).
[CrossRef]

1964 (1)

1960 (1)

M. J. Levin, “Generation of a sampled Gaussian time series having a specified correlation function,” IRE Trans. Inf. Theory IT-6, 545–548 (1960);J. C. Camparo, P. Lambropoulos, “Monte Carlo simulations of field fluctuations in strongly driven resonant transitions,” Phys. Rev. A 47, 480–494 (1993).
[CrossRef] [PubMed]

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, (Dover, New York, 1972), Chap. 25.

Beland, R. R.

R. R. Beland, “Propagation through atmospheric optical turbulence,” in Atmospheric Propagation of Radiation, F. G. Smith, ed. (SPIE Press, Bellingham, Wash., 1993), Chap. 2.

Bracher, C.

Bufton, J.

J. Bufton, S. H. Genatt, “Simultaneous observations of atmospheric turbulence effects on stellar irradiance and phase,” Astron. J. 76, 378–386 (1971); see also G. Parry, J. G. Walker, R. J. Scaddan, “On the statistics of stellar speckle patterns and pupil plane scintillation,” Opt. Acta 26, 563–574 (1979).
[CrossRef]

Churnside, J. H.

Clifford, S. F.

Cochetti, F.

Consortini, A.

Dravins, D.

D. Dravins, L. Lindegren, E. Mezey, A. T. Young, “Atmospheric intensity scintillation of stars. I. Statistical distributions and temporal properties,” Publ. Astron. Soc. Pac. 109, 173–207 (1997).
[CrossRef]

Flatté, S. M.

Fried, D. L.

The author received the text of D. Fried’s phase-screen generation algorithm from D. L. Fried, 14671 Tumbleweed Lane, Monterey County, Calif. 95076 (private communication).

Fröberg, C-E.

C-E. Fröberg, Introduction to Numerical Analysis (Addison–Wesley, Reading, Mass., 1965), Chap. 10.

Genatt, S. H.

J. Bufton, S. H. Genatt, “Simultaneous observations of atmospheric turbulence effects on stellar irradiance and phase,” Astron. J. 76, 378–386 (1971); see also G. Parry, J. G. Walker, R. J. Scaddan, “On the statistics of stellar speckle patterns and pupil plane scintillation,” Opt. Acta 26, 563–574 (1979).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw–Hill, San Francisco, Calif., 1968).

Gracheva, M. E.

M. E. Gracheva, A. S. Gurvich, S. S. Kashkarov, Vl. V. Pokasov, “Similarity relations and their experimental verification for strong intensity fluctuations of laser radiation,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, Berlin, 1978), Chap. 4.

Gurvich, A. S.

M. E. Gracheva, A. S. Gurvich, S. S. Kashkarov, Vl. V. Pokasov, “Similarity relations and their experimental verification for strong intensity fluctuations of laser radiation,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, Berlin, 1978), Chap. 4.

Hays, William L.

William L. Hays, Statistics (Holt, Rinehart & Winston, New York, 1988), Chap. 5.

Hill, R. J.

Hufnagel, R. E.

Kashkarov, S. S.

M. E. Gracheva, A. S. Gurvich, S. S. Kashkarov, Vl. V. Pokasov, “Similarity relations and their experimental verification for strong intensity fluctuations of laser radiation,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, Berlin, 1978), Chap. 4.

Knepp, D. L.

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

Lawrence, R. S.

Levin, M. J.

M. J. Levin, “Generation of a sampled Gaussian time series having a specified correlation function,” IRE Trans. Inf. Theory IT-6, 545–548 (1960);J. C. Camparo, P. Lambropoulos, “Monte Carlo simulations of field fluctuations in strongly driven resonant transitions,” Phys. Rev. A 47, 480–494 (1993).
[CrossRef] [PubMed]

Lindegren, L.

D. Dravins, L. Lindegren, E. Mezey, A. T. Young, “Atmospheric intensity scintillation of stars. I. Statistical distributions and temporal properties,” Publ. Astron. Soc. Pac. 109, 173–207 (1997).
[CrossRef]

Martin, J.

Martin, J. M.

Mezey, E.

D. Dravins, L. Lindegren, E. Mezey, A. T. Young, “Atmospheric intensity scintillation of stars. I. Statistical distributions and temporal properties,” Publ. Astron. Soc. Pac. 109, 173–207 (1997).
[CrossRef]

Ochs, G. R.

Pokasov, Vl. V.

M. E. Gracheva, A. S. Gurvich, S. S. Kashkarov, Vl. V. Pokasov, “Similarity relations and their experimental verification for strong intensity fluctuations of laser radiation,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, Berlin, 1978), Chap. 4.

Roggemann, M. C.

Stanley, N. R.

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, (Dover, New York, 1972), Chap. 25.

Strohbehn, J. W.

J. W. Strohbehn, “Modern theories in the propagation of optical waves in a turbulent medium,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn ed. (Springer–Verlag, Berlin, 1978), Chap. 3.

Tatarski, V. I.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961), Chap. 8.

Troxel, S. E.

Wang, G.-Y.

Welsh, R. M.

Whale, H. A.

H. A. Whale, “Diffraction of a plane wave by a random phase screen,” J. Atmos. Terr. Phys. 35, 263–274 (1973);R. Buckley, “Diffraction by a random phase-changing screen: a numerical experiment,” J. Atmos. Terr. Phys. 37, 1431–1446 (1975);W. P. Brown, “Computer simulation of adaptive optical systems,” Hughes Research Laboratories Report, contract N60921–74–C–0249, 1975; B. L. McGlamery, “Computer simulation studies of compensation of turbulence degraded images,” in Image Processing, J. C. Urbach, ed., Proc. SPIE74, 225–233 (1976)
[CrossRef]

Young, A. T.

D. Dravins, L. Lindegren, E. Mezey, A. T. Young, “Atmospheric intensity scintillation of stars. I. Statistical distributions and temporal properties,” Publ. Astron. Soc. Pac. 109, 173–207 (1997).
[CrossRef]

Yura, H. T.

Appl. Opt. (1)

Astron. J. (1)

J. Bufton, S. H. Genatt, “Simultaneous observations of atmospheric turbulence effects on stellar irradiance and phase,” Astron. J. 76, 378–386 (1971); see also G. Parry, J. G. Walker, R. J. Scaddan, “On the statistics of stellar speckle patterns and pupil plane scintillation,” Opt. Acta 26, 563–574 (1979).
[CrossRef]

IRE Trans. Inf. Theory (1)

M. J. Levin, “Generation of a sampled Gaussian time series having a specified correlation function,” IRE Trans. Inf. Theory IT-6, 545–548 (1960);J. C. Camparo, P. Lambropoulos, “Monte Carlo simulations of field fluctuations in strongly driven resonant transitions,” Phys. Rev. A 47, 480–494 (1993).
[CrossRef] [PubMed]

J. Atmos. Terr. Phys. (1)

H. A. Whale, “Diffraction of a plane wave by a random phase screen,” J. Atmos. Terr. Phys. 35, 263–274 (1973);R. Buckley, “Diffraction by a random phase-changing screen: a numerical experiment,” J. Atmos. Terr. Phys. 37, 1431–1446 (1975);W. P. Brown, “Computer simulation of adaptive optical systems,” Hughes Research Laboratories Report, contract N60921–74–C–0249, 1975; B. L. McGlamery, “Computer simulation studies of compensation of turbulence degraded images,” in Image Processing, J. C. Urbach, ed., Proc. SPIE74, 225–233 (1976)
[CrossRef]

J. Opt. Soc. Am. (3)

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

Proc. IEEE (1)

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

Publ. Astron. Soc. Pac. (1)

D. Dravins, L. Lindegren, E. Mezey, A. T. Young, “Atmospheric intensity scintillation of stars. I. Statistical distributions and temporal properties,” Publ. Astron. Soc. Pac. 109, 173–207 (1997).
[CrossRef]

Other (12)

In the following, we refer to the lower 9 km of atmosphere as the troposphere, the 9–12-km region as the tropopause, and the 12–20-km region as the lower stratosphere.

R. R. Beland, “Propagation through atmospheric optical turbulence,” in Atmospheric Propagation of Radiation, F. G. Smith, ed. (SPIE Press, Bellingham, Wash., 1993), Chap. 2.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961), Chap. 8.

M. E. Gracheva, A. S. Gurvich, S. S. Kashkarov, Vl. V. Pokasov, “Similarity relations and their experimental verification for strong intensity fluctuations of laser radiation,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer-Verlag, Berlin, 1978), Chap. 4.

C-E. Fröberg, Introduction to Numerical Analysis (Addison–Wesley, Reading, Mass., 1965), Chap. 10.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions, (Dover, New York, 1972), Chap. 25.

The author received the text of D. Fried’s phase-screen generation algorithm from D. L. Fried, 14671 Tumbleweed Lane, Monterey County, Calif. 95076 (private communication).

J. W. Goodman, Introduction to Fourier Optics (McGraw–Hill, San Francisco, Calif., 1968).

For completeness, we note that the pixel size of ∼0.3 cm–0.4 cm introduces an inner scale of turbulence into the simulations. This is an unavoidable consequence of all finite element numerical simulations.

In principle, a telescope of any diameter may be simulated. However, for larger-aperture telescopes to be considered, the phase-screen length must also become larger, which in turn requires a larger number of pixels to span the screen’s length. (As discussed in Appendix A, the pixel size is bounded by considerations related to the strength of turbulence and the interscreen spacing.) The computational requirements for dealing with phase screens that have large numbers of pixels can quickly become prohibitive.

William L. Hays, Statistics (Holt, Rinehart & Winston, New York, 1988), Chap. 5.

J. W. Strohbehn, “Modern theories in the propagation of optical waves in a turbulent medium,” in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn ed. (Springer–Verlag, Berlin, 1978), Chap. 3.

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Figures (6)

Fig. 1
Fig. 1

Cn2(h)h5/6 versus altitude for the Hufnagel–Valley model of atmospheric turbulence. For the HV-21 profile, A=1.7×10-14 m-2/3, vs=21 m/s, and the strength of turbulence is varied by changing Γ (for the graph Γ=3.85). For the high-altitude profile, labeled HA, A=1.0×10-13 m-2/3, Γ=1, and the strength of turbulence is varied by changing the rms wind speed (for the graph vs=47 m/s). Each profile shown in the figure yields σR2=0.5.

Fig. 2
Fig. 2

Circles, log-amplitude correlation function for the HV-21 model with Γ=1 (i.e., weak turbulence): ro=11.8 cm, and σR2=0.11. Solid curve, prediction from Rytov’s theory with Eq. (4).

Fig. 3
Fig. 3

Phase-screen calculation of irradiance variance σI2 versus the Rytov standard deviation σR. Open circles, HV-21 model; solid circles, HA model. Dashed curve, Rytov theory [i.e., expression (1)]; solid curves, Eq. (5) with σo2=1.6 and σo2=2.5 for the HA and the HV-21 profiles, respectively.

Fig. 4
Fig. 4

Phase-screen calculation of the aperture-averaged irradiance variance σI2 versus the Rytov standard deviation σR. Open circles, HV-21 model; solid circles, HA model. The telescope diameter in the simulations was 25 cm.

Fig. 5
Fig. 5

Log-amplitude correlation function Bχ(ρ)/Bχ(0) in the strong-focusing regime (i.e., σR=4.8): open circles, HV-21 profile; solid circles, HA profile.

Fig. 6
Fig. 6

Rytov variance parameterized by the scintillation parameter f as discussed in the text. Open circles, HV-21 profile with f averaged over the tropopause; solid circles, HA profile with f averaged over the tropopause; dashed curve, HV-21 profile with f averaged over the lower troposphere. For the tropopause-averaged curves, the HV-21 and the HA profile Rytov variances at f=1 are in reasonable agreement with their respective σo2 values.

Tables (1)

Tables Icon

Table 1 Altitudes of Phase Screens

Equations (13)

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σI2σR2exp2.25k7/6sec11/6(ϕ)0Cn2(h)h5/6dh-1.
Cn2(h)=Γ{8.15×10-56vs2h10exp[-(h/1000)]+2.7×10-16exp[-(h/1500)]+A exp[-(h/100)]}.
E(ρ, hi-)=Eo(ρ, hi-)exp[iϕ(ρ, hi-)],
Bχ(ρ)=4π2k20HCn2(h)(0.033)0Jo(κρ)κ(-11/3)×sin2κ2h2kκdκdh.
σI2=σo2[1-exp[-(σR2/σo2)]].
ρo(h)=1.45 k2hCn2(h)dh-3/5.
f1h2-h1h1h2ρo(h)λhdh,
0HCn2(h)dh=00.3 kmCn2(h)dh+0.3 kmHCn2(h)dh=a1Cn2(h1)+i=2NsaiCn2(hi).
ro,i-5/3=0.423k2aiCn2(hi).
θ(m, hi)=θTTE(m)-βm-M2I+γgm-M2I.
β=12M2(M2-1)m=1Mm-(M+1)2IθTTE(m).
ψ(h2)=ψ(h1) exp-iκ2(h1-h2)2k,
M2π2Δz2kL2-(M-1)2π2Δz2kL2<π,

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