Abstract

In this Letter we have analyzed the temporal correlations of the angle-of-arrival fluctuations of stellar images. Experimentally measured data were carefully examined by implementing multifractal detrended fluctuation analysis. This algorithm is able to discriminate the presence of fractal and multifractal structures in recorded time sequences. We have confirmed that turbulence-degraded stellar wavefronts are compatible with a long-memory correlated monofractal process. This experimental result is quite significant for the accurate comprehension and modeling of the atmospheric turbulence effects on the stellar images. It can also be of great utility within the adaptive optics field.

© 2014 Optical Society of America

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References

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2013

D. Grech and G. Pamuła, Phys. A 392, 5845 (2013).
[CrossRef]

2012

E. A. F. Ihlen, Front. Physiol. 3, 141 (2012).
[CrossRef]

A. Ziad, J. Borgnino, W. Dali Ali, A. Berdja, J. Maire, and F. Martin, J. Opt. 14, 045705 (2012).
[CrossRef]

2011

A. Y. Schumann and J. W. Kantelhardt, Phys. A 390, 2637 (2011).
[CrossRef]

2010

2006

P. Oświęcimka, J. Kwapień, and S. Drożdż, Phys. Rev. E 74, 016103 (2006).
[CrossRef]

2002

J. W. Kantelhardt, S. A. Zschiegner, E. Koscielny-Bunde, S. Havlin, A. Bunde, and H. E. Stanley, Phy. A 316, 87 (2002).
[CrossRef]

2001

J. W. Kantelhardt, E. Koscielny-Bunde, H. H. A. Rego, S. Havlin, and A. Bunde, Phys. A 295, 441 (2001).
[CrossRef]

2000

1997

1996

P. Abry and F. Sellan, Appl. Comput. Harmon. Anal. 3, 377 (1996).
[CrossRef]

1995

1994

C.-K. Peng, S. V. Buldyrev, S. Havlin, M. Simons, H. E. Stanley, and A. L. Goldberger, Phys. Rev. E 49, 1685 (1994).
[CrossRef]

C. Schwartz, G. Baum, and E. N. Ribak, J. Opt. Soc. Am. A 11, 444 (1994).
[CrossRef]

1992

1991

1988

I. Freund, M. Rosenbluh, and S. Feng, Phys. Rev. Lett. 61, 2328 (1988).
[CrossRef]

1968

B. B. Mandelbrot and J. W. Van Ness, SIAM Rev. 10, 422 (1968).
[CrossRef]

Abry, P.

P. Abry and F. Sellan, Appl. Comput. Harmon. Anal. 3, 377 (1996).
[CrossRef]

Aitken, G. J. M.

Baum, G.

Berdja, A.

A. Ziad, J. Borgnino, W. Dali Ali, A. Berdja, J. Maire, and F. Martin, J. Opt. 14, 045705 (2012).
[CrossRef]

Boreman, G. D.

Borgnino, J.

A. Ziad, J. Borgnino, W. Dali Ali, A. Berdja, J. Maire, and F. Martin, J. Opt. 14, 045705 (2012).
[CrossRef]

A. Ziad, R. Conan, A. Tokovinin, F. Martin, and J. Borgnino, Appl. Opt. 39, 5415 (2000).
[CrossRef]

Buldyrev, S. V.

C.-K. Peng, S. V. Buldyrev, S. Havlin, M. Simons, H. E. Stanley, and A. L. Goldberger, Phys. Rev. E 49, 1685 (1994).
[CrossRef]

Bunde, A.

J. W. Kantelhardt, S. A. Zschiegner, E. Koscielny-Bunde, S. Havlin, A. Bunde, and H. E. Stanley, Phy. A 316, 87 (2002).
[CrossRef]

J. W. Kantelhardt, E. Koscielny-Bunde, H. H. A. Rego, S. Havlin, and A. Bunde, Phys. A 295, 441 (2001).
[CrossRef]

Conan, J.-M.

Conan, R.

Dainty, J. C.

Dali Ali, W.

A. Ziad, J. Borgnino, W. Dali Ali, A. Berdja, J. Maire, and F. Martin, J. Opt. 14, 045705 (2012).
[CrossRef]

Dayton, D.

Drozdz, S.

P. Oświęcimka, J. Kwapień, and S. Drożdż, Phys. Rev. E 74, 016103 (2006).
[CrossRef]

Du, W.

Feder, J.

J. Feder, Fractals (Plenum, 1988).

Feng, S.

I. Freund, M. Rosenbluh, and S. Feng, Phys. Rev. Lett. 61, 2328 (1988).
[CrossRef]

Freund, I.

I. Freund, M. Rosenbluh, and S. Feng, Phys. Rev. Lett. 61, 2328 (1988).
[CrossRef]

Goldberger, A. L.

C.-K. Peng, S. V. Buldyrev, S. Havlin, M. Simons, H. E. Stanley, and A. L. Goldberger, Phys. Rev. E 49, 1685 (1994).
[CrossRef]

Gonglewski, J.

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, 1985).

Grech, D.

D. Grech and G. Pamuła, Phys. A 392, 5845 (2013).
[CrossRef]

Havlin, S.

J. W. Kantelhardt, S. A. Zschiegner, E. Koscielny-Bunde, S. Havlin, A. Bunde, and H. E. Stanley, Phy. A 316, 87 (2002).
[CrossRef]

J. W. Kantelhardt, E. Koscielny-Bunde, H. H. A. Rego, S. Havlin, and A. Bunde, Phys. A 295, 441 (2001).
[CrossRef]

C.-K. Peng, S. V. Buldyrev, S. Havlin, M. Simons, H. E. Stanley, and A. L. Goldberger, Phys. Rev. E 49, 1685 (1994).
[CrossRef]

Hege, E. K.

Ihlen, E. A. F.

E. A. F. Ihlen, Front. Physiol. 3, 141 (2012).
[CrossRef]

Jiang, Y.

Jorgenson, M. B.

Kantelhardt, J. W.

A. Y. Schumann and J. W. Kantelhardt, Phys. A 390, 2637 (2011).
[CrossRef]

J. W. Kantelhardt, S. A. Zschiegner, E. Koscielny-Bunde, S. Havlin, A. Bunde, and H. E. Stanley, Phy. A 316, 87 (2002).
[CrossRef]

J. W. Kantelhardt, E. Koscielny-Bunde, H. H. A. Rego, S. Havlin, and A. Bunde, Phys. A 295, 441 (2001).
[CrossRef]

Koscielny-Bunde, E.

J. W. Kantelhardt, S. A. Zschiegner, E. Koscielny-Bunde, S. Havlin, A. Bunde, and H. E. Stanley, Phy. A 316, 87 (2002).
[CrossRef]

J. W. Kantelhardt, E. Koscielny-Bunde, H. H. A. Rego, S. Havlin, and A. Bunde, Phys. A 295, 441 (2001).
[CrossRef]

Kwapien, J.

P. Oświęcimka, J. Kwapień, and S. Drożdż, Phys. Rev. E 74, 016103 (2006).
[CrossRef]

Ma, J.

Madec, P.-Y.

Maire, J.

A. Ziad, J. Borgnino, W. Dali Ali, A. Berdja, J. Maire, and F. Martin, J. Opt. 14, 045705 (2012).
[CrossRef]

Mandelbrot, B. B.

B. B. Mandelbrot and J. W. Van Ness, SIAM Rev. 10, 422 (1968).
[CrossRef]

Martin, F.

A. Ziad, J. Borgnino, W. Dali Ali, A. Berdja, J. Maire, and F. Martin, J. Opt. 14, 045705 (2012).
[CrossRef]

A. Ziad, R. Conan, A. Tokovinin, F. Martin, and J. Borgnino, Appl. Opt. 39, 5415 (2000).
[CrossRef]

McGaughey, D. R.

Nicholls, T. W.

Oswiecimka, P.

P. Oświęcimka, J. Kwapień, and S. Drożdż, Phys. Rev. E 74, 016103 (2006).
[CrossRef]

Pamula, G.

D. Grech and G. Pamuła, Phys. A 392, 5845 (2013).
[CrossRef]

Peng, C.-K.

C.-K. Peng, S. V. Buldyrev, S. Havlin, M. Simons, H. E. Stanley, and A. L. Goldberger, Phys. Rev. E 49, 1685 (1994).
[CrossRef]

Pierson, B.

Rego, H. H. A.

J. W. Kantelhardt, E. Koscielny-Bunde, H. H. A. Rego, S. Havlin, and A. Bunde, Phys. A 295, 441 (2001).
[CrossRef]

Ribak, E. N.

Roggemann, M. C.

M. C. Roggemann and B. Welsh, Imaging Through Turbulence (CRC Press, 1996).

Rosenbluh, M.

I. Freund, M. Rosenbluh, and S. Feng, Phys. Rev. Lett. 61, 2328 (1988).
[CrossRef]

Rousset, G.

Schumann, A. Y.

A. Y. Schumann and J. W. Kantelhardt, Phys. A 390, 2637 (2011).
[CrossRef]

Schwartz, C.

Sellan, F.

P. Abry and F. Sellan, Appl. Comput. Harmon. Anal. 3, 377 (1996).
[CrossRef]

Simons, M.

C.-K. Peng, S. V. Buldyrev, S. Havlin, M. Simons, H. E. Stanley, and A. L. Goldberger, Phys. Rev. E 49, 1685 (1994).
[CrossRef]

Spielbusch, B.

Stanley, H. E.

J. W. Kantelhardt, S. A. Zschiegner, E. Koscielny-Bunde, S. Havlin, A. Bunde, and H. E. Stanley, Phy. A 316, 87 (2002).
[CrossRef]

C.-K. Peng, S. V. Buldyrev, S. Havlin, M. Simons, H. E. Stanley, and A. L. Goldberger, Phys. Rev. E 49, 1685 (1994).
[CrossRef]

Tan, L.

Tokovinin, A.

Van Ness, J. W.

B. B. Mandelbrot and J. W. Van Ness, SIAM Rev. 10, 422 (1968).
[CrossRef]

Welsh, B.

M. C. Roggemann and B. Welsh, Imaging Through Turbulence (CRC Press, 1996).

Ziad, A.

A. Ziad, J. Borgnino, W. Dali Ali, A. Berdja, J. Maire, and F. Martin, J. Opt. 14, 045705 (2012).
[CrossRef]

A. Ziad, R. Conan, A. Tokovinin, F. Martin, and J. Borgnino, Appl. Opt. 39, 5415 (2000).
[CrossRef]

Zschiegner, S. A.

J. W. Kantelhardt, S. A. Zschiegner, E. Koscielny-Bunde, S. Havlin, A. Bunde, and H. E. Stanley, Phy. A 316, 87 (2002).
[CrossRef]

Appl. Comput. Harmon. Anal.

P. Abry and F. Sellan, Appl. Comput. Harmon. Anal. 3, 377 (1996).
[CrossRef]

Appl. Opt.

Front. Physiol.

E. A. F. Ihlen, Front. Physiol. 3, 141 (2012).
[CrossRef]

J. Opt.

A. Ziad, J. Borgnino, W. Dali Ali, A. Berdja, J. Maire, and F. Martin, J. Opt. 14, 045705 (2012).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Express

Opt. Lett.

Phy. A

J. W. Kantelhardt, S. A. Zschiegner, E. Koscielny-Bunde, S. Havlin, A. Bunde, and H. E. Stanley, Phy. A 316, 87 (2002).
[CrossRef]

Phys. A

J. W. Kantelhardt, E. Koscielny-Bunde, H. H. A. Rego, S. Havlin, and A. Bunde, Phys. A 295, 441 (2001).
[CrossRef]

D. Grech and G. Pamuła, Phys. A 392, 5845 (2013).
[CrossRef]

A. Y. Schumann and J. W. Kantelhardt, Phys. A 390, 2637 (2011).
[CrossRef]

Phys. Rev. E

P. Oświęcimka, J. Kwapień, and S. Drożdż, Phys. Rev. E 74, 016103 (2006).
[CrossRef]

C.-K. Peng, S. V. Buldyrev, S. Havlin, M. Simons, H. E. Stanley, and A. L. Goldberger, Phys. Rev. E 49, 1685 (1994).
[CrossRef]

Phys. Rev. Lett.

I. Freund, M. Rosenbluh, and S. Feng, Phys. Rev. Lett. 61, 2328 (1988).
[CrossRef]

SIAM Rev.

B. B. Mandelbrot and J. W. Van Ness, SIAM Rev. 10, 422 (1968).
[CrossRef]

Other

J. Feder, Fractals (Plenum, 1988).

J. W. Goodman, Statistical Optics (Wiley, 1985).

M. C. Roggemann and B. Welsh, Imaging Through Turbulence (CRC Press, 1996).

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Figures (5)

Fig. 1.
Fig. 1.

Representative sample of the AA fluctuations (top) and average PSD of the 19 sequences of real wavefront slopes (bottom). The theoretical expected 2/3 power-law behavior at the low-frequency regime is plotted (red-dashed line). The knee frequency is also indicated (vertical black-dashed line). Peaks observed at high frequencies are due to vibrations in the experimental arrangement.

Fig. 2.
Fig. 2.

Fluctuation functions Fq(s) as a function of the scale s for the AA fluctuations plotted in Fig. 1. A detrending polynomial of order m=2 and 100 different scales s[10,N/4] equally spaced in the logarithmic scale were employed in the MF-DFA implementation. The order q (q=20,19,,20) increases from bottom to top. The behavior observed is representative for the whole data set.

Fig. 3.
Fig. 3.

Fluctuation functions F2(s) as a function of the scale s for the 19 independent sets of AA fluctuations. The slope of the best linear fit obtained for each one of these fluctuation functions is the Hurst exponent estimator (standard DFA technique [12]) of the experimental records.

Fig. 4.
Fig. 4.

Generalized Hurst exponents h(q), estimated in the full fitting range s[10,N/4], as a function of the order q for the 19 independent sets of AA fluctuations. The vertical black continuous line indicates the estimated values for the main Hurst exponent (H=h(2)).

Fig. 5.
Fig. 5.

(a) Estimated values for the Hurst exponent H (blue dots) and multifractality degree Δh (green dots) for the nineteen independent experimental sets of AA fluctuations. (b) Related boxplots for both quantifiers. The theoretical expected value for the Hurst exponent within the Kolmogorov model (H=5/6) is indicated (horizontal blue continuous lines).

Equations (6)

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

Y(i)=t=1i(xtx),
x=(t=1Nxt)/N,
Fm2(ν,s)=1si=1s{Y[(ν1)s+i]yν,m(i)}2,
Fq(s)={1N/sν=1N/s[Fm2(ν,s)]q/2}1/q,
Fq(s)sh(q)
Δhh(q)h(q),

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