Abstract

Experimental results indicate that the statistics of phase measured across a telescope aperture do not always obey the power laws associated with the Kolmogorov model of atmospheric turbulence. We show that the statistical relations between a wave front and its aperture-averaged first derivative previously derived for a Kolmogorov spectrum can be easily generalized for any power law. We also show that a Shack–Hartmann sensor can be used to measure the form of the structure function of phase fluctuations, and experimental data are presented.

© 1995 Optical Society of America

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References

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  1. V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961) (translated from Russian by R. A. Silverman).
  2. D. L. Fried, J. Opt. Soc. Am. 55, 1427 (1965).
    [CrossRef]
  3. D. Dayton, B. Pierson, B. Spielbusch, J. Gonglewski, Opt. Lett. 17, 1737 (1992).
    [CrossRef] [PubMed]
  4. M. Bester, W. C. Danchi, L. J. Degiacomi, C. H. Townes, Astrophys. J. 392, 357 (1992).
    [CrossRef]
  5. R. G. Buser, J. Opt. Soc. Am. 61, 488 (1971).
    [CrossRef]
  6. G. D. Boreman, J. C. Dainty, “Zernike expansions for non-Kolmogorov turbulence,” J. Opt. Soc. Am. A (to be published).
  7. F. Roddier, Prog. Opt. 19, 281 (1981).
    [CrossRef]
  8. M. Sarazin, F. Roddier, Astron. Astrophys. 227, 294 (1990).
  9. D. L. Fried, Radio Sci. 10(1), 71 (1975).
    [CrossRef]

1992 (2)

D. Dayton, B. Pierson, B. Spielbusch, J. Gonglewski, Opt. Lett. 17, 1737 (1992).
[CrossRef] [PubMed]

M. Bester, W. C. Danchi, L. J. Degiacomi, C. H. Townes, Astrophys. J. 392, 357 (1992).
[CrossRef]

1990 (1)

M. Sarazin, F. Roddier, Astron. Astrophys. 227, 294 (1990).

1981 (1)

F. Roddier, Prog. Opt. 19, 281 (1981).
[CrossRef]

1975 (1)

D. L. Fried, Radio Sci. 10(1), 71 (1975).
[CrossRef]

1971 (1)

1965 (1)

Bester, M.

M. Bester, W. C. Danchi, L. J. Degiacomi, C. H. Townes, Astrophys. J. 392, 357 (1992).
[CrossRef]

Boreman, G. D.

G. D. Boreman, J. C. Dainty, “Zernike expansions for non-Kolmogorov turbulence,” J. Opt. Soc. Am. A (to be published).

Buser, R. G.

Dainty, J. C.

G. D. Boreman, J. C. Dainty, “Zernike expansions for non-Kolmogorov turbulence,” J. Opt. Soc. Am. A (to be published).

Danchi, W. C.

M. Bester, W. C. Danchi, L. J. Degiacomi, C. H. Townes, Astrophys. J. 392, 357 (1992).
[CrossRef]

Dayton, D.

Degiacomi, L. J.

M. Bester, W. C. Danchi, L. J. Degiacomi, C. H. Townes, Astrophys. J. 392, 357 (1992).
[CrossRef]

Fried, D. L.

Gonglewski, J.

Pierson, B.

Roddier, F.

M. Sarazin, F. Roddier, Astron. Astrophys. 227, 294 (1990).

F. Roddier, Prog. Opt. 19, 281 (1981).
[CrossRef]

Sarazin, M.

M. Sarazin, F. Roddier, Astron. Astrophys. 227, 294 (1990).

Spielbusch, B.

Tatarski, V. I.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961) (translated from Russian by R. A. Silverman).

Townes, C. H.

M. Bester, W. C. Danchi, L. J. Degiacomi, C. H. Townes, Astrophys. J. 392, 357 (1992).
[CrossRef]

Astron. Astrophys. (1)

M. Sarazin, F. Roddier, Astron. Astrophys. 227, 294 (1990).

Astrophys. J. (1)

M. Bester, W. C. Danchi, L. J. Degiacomi, C. H. Townes, Astrophys. J. 392, 357 (1992).
[CrossRef]

J. Opt. Soc. Am. (2)

Opt. Lett. (1)

Prog. Opt. (1)

F. Roddier, Prog. Opt. 19, 281 (1981).
[CrossRef]

Radio Sci. (1)

D. L. Fried, Radio Sci. 10(1), 71 (1975).
[CrossRef]

Other (2)

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961) (translated from Russian by R. A. Silverman).

G. D. Boreman, J. C. Dainty, “Zernike expansions for non-Kolmogorov turbulence,” J. Opt. Soc. Am. A (to be published).

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Figures (4)

Fig. 1
Fig. 1

Dependence of the ratio of transverse-to-longitudinal differential angle-of-arrival variance on the exponent (β) of the power spectrum of the phase. Fried’s calculated values for a Kolmogorov spectrum are shown by the dotted curve; Sarazin and Roddier’s approximate formula is plotted as the dashed curve.

Fig. 2
Fig. 2

Comparison of simulated centroid motions (symbols) with theory (solid curves).

Fig. 3
Fig. 3

Ratio of differential variances found in experimental data taken at La Palma (symbols) compared with theoretical curves for varying β.

Fig. 4
Fig. 4

Ratio of differential variances found in experimental data taken at Calar Alto (symbols) compared with theoretical curves for varying β.

Equations (12)

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Φ φ ( k ) = 0.023 k - 11 / 3 r 0 5 / 3 .
Φ φ ( k ) = A β k - β r 0 β - 2 .
D φ ( r ) = φ ( r ) - φ ( r + r ) 2 ,
D φ ( r ) = γ β ( r ρ 0 ) β - 2 .
α ( x , y ) = - ( λ 2 π ) x φ ( x , y ) ,
B α ( μ , η ) = α ( x , y ) α ( x + μ , y + η ) .
B α ( μ , η ) = - λ 2 8 π 2 2 μ 2 D φ ( μ , η ) .
D φ ( μ , η ) = γ β ρ 0 - β + 2 ( μ 2 + η 2 ) ( β - 2 ) / 2 .
B α ( d , 0 ) = 0.0127 γ β λ 2 ρ 0 - β + 2 ( β - 2 ) ( β - 3 ) d β - 4 ,
B α ( 0 , d ) = 0.0127 γ β λ 2 ρ 0 - β + 2 ( β - 2 ) d β - 4 .
B α ( d , 0 ) B α ( 0 , d ) = β - 3.
( α 1 - α 2 ) 2 = ( 4 π ) 4 ( λ D ) 2 0 2 π d θ 0 1 u d u { 1 8 cos - 1 u + ( 1 - u 2 ) 1 / 2 [ ( u 3 12 - 5 u 24 ) + ( u 3 3 - u 3 ) cos 2 θ ] } × ( 1 2 D φ { D [ S 2 + 2 S u cos ( θ + ψ ) + u 2 ] 1 / 2 } + 1 2 D φ { D [ S 2 - 2 S u cos ( θ + ψ ) + u 2 ] 1 / 2 } - D φ ( D u ) ) .

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