Abstract

We present the results of a laser beam passing through a turbulent medium. First we measure the geometric parameters and the spatial coherence of the beam as a function of wind velocities. A multifractal detrended fluctuation analysis algorithm is applied to determine the multifractal scaling behavior of the intensity patterns. The measurements are fitted with models used in the analysis of river runoff records. We show the surprising result that the multifractality decreases when the spatial coherence of the laser is decreased. Universal scaling properties could be applied to the spatial characterization of a laser propagating in a turbulent medium or random medium.

© 2006 Optical Society of America

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  1. P. J. Gardner, M. C. Roggemann, B. M. Welsh, R. D. Bowersox, and T. E. Luke, "Statistical anisotropy in free turbulence for mixing layers at high Reynolds numbers," Appl. Opt. 35, 4879-4889 (1996).
    [CrossRef] [PubMed]
  2. H. Okayama and L. Z. Wang, "Measurement of the spatial coherence of light influenced by turbulence," Appl. Opt. 38, 2342-2345 (1999).
    [CrossRef]
  3. M. S. Belenkii and V. L. Mironov, "Coherence of the field of a laser beam in a turbulent atmosphere," Sov. J. Quantum Electron. 10, 595-597 (1980).
    [CrossRef]
  4. Z. I. Feizulin and Yu. A. Kravtsov, "Broadening of a laser beam in a turbulent medium," Radio Quantum Electron. 10, 33-35 (1967).
    [CrossRef]
  5. L. C. Andrews, W. B. Miller, and J. C. Ricklin, "Spatial coherence of a Gaussian-beam wave in weak and strong optical turbulence," J. Opt. Soc. Am. A 11, 1653-1660 (1994).
    [CrossRef]
  6. J. W. Kantelhardt, S. A. Zschiegner, E. Koscielny-Bunde, A. Bunde, S. Havlin, and H. E. Stanley, "Multifractal detrended fluctuations analysis of nonstationary time series," Physica A 316, 87-114 (2002).
    [CrossRef]
  7. R. F. Voss, "Random fractals self-affinity in noise, music mountains, and clouds," Physica D 38, 362-371 (1989).
    [CrossRef]
  8. C. K. Peng, S. V. Buldyrev, S. Havlin, M. Simons, H. E. Stanley, and A. L. Goldberger, "Mosaic organinazation of DNA nucleotides," Phys. Rev. E. 49, 1685-1689 (1994).
    [CrossRef]
  9. P. Ch. Ivanov, L. A. Nunes Amaral, S. Havlin, M. G. Rosenblum, H. E. Stanley, and Z. R. Struzik, "From 1/f noise to multifractal cascades in heartbeat dynamics," Chaos 11, 641-652 (2001).
    [CrossRef]
  10. E. Koscielny-Bunde, A. Bunde, S. Havlin, H. E. Roman, Y. Goldreich, and H.-J. Schellnhuber, "Indication of a universal persistence law governing atmospheric variability," Phys. Rev. Lett. 81, 729-732 (1998).
    [CrossRef]
  11. R. B. Govindan, D. Vyushin, A. Bunde, S. Brenner, S. Havlin, and H. J. Schellnhuber, "Global climate models violate scaling of the observed atmospheric variability," Phys. Rev. Lett. 89, 0285011-0285014 (2002).
    [CrossRef]
  12. Y. Liu, P. Gopikrishnan, P. Cizeau, M. Meyer, C.-K. Peng, and H. E. Stanley, "Statistical properties of the volatility of price fluctuations," Phys. Rev. E 60, 1390-1400 (1999).
    [CrossRef]
  13. E. Koscielny-Bunde, J. W. Kantelhardt, P. Braun, A. Bunde, and S. Havlin, "Long-term persistence and multifractality of river runoff records: detrended fluctuation studies," arXiv.org e-Print archive, physics/0305078, 19 May 2003, http://pbparxiv.org/abs/physics/0305078.
  14. F. Schmitt, D. Schertzer, S. Lovejoy, and Y. Brunet, "Empirical study of multifractal phase transitions in atmospheric turbulence," Nonlinear Processes Geophys. 1, 95-100 (1994).
    [CrossRef]
  15. S. Lovejoy, D. Schertzer, and J. D. Stanway, "Direct evidence of multifractal atmospheric cascades from planetary scales down to 1 km," Phys. Rev. Lett. 86, 5200-5204 (2001).
    [CrossRef] [PubMed]
  16. C. Innocenti and A. Consortini, "Estimate of characteristic scale of atmospheric turbulence by thin beams: comparison between the von Karman and Hill-Andrez models," J. Mod. Opt. 51, 333-342 (2003).
    [CrossRef]
  17. A. Consortini, Y. Y. Sun, C. Innocenti, and Z. P. Li, "Measuring inner scale of atmospheric turbulence by angle of arrival and scintillation," Opt. Commun. 216, 19-23 (2003).
    [CrossRef]
  18. W. Pratt, Digital Image Processing (Wiley, 1991), pp. 630-634.
  19. L. A. N. Amaral, P. Ch. Ivanov, N. Aoyagi, I. Hidaka, S. Tomono, A. L. Goldberger, H. E. Stanley, and Y. Yamamoto, "Behavioral-independent features of complex heartbeat dynamics," Phys. Rev. Lett. 86, 6026-6029 (2001).
    [CrossRef]
  20. N. Scafetta, L. Griffin, and B. J. West, "Hölder exponent spectra for human gait," Physica A 328, 561-583 (2003).
    [CrossRef]

2003

C. Innocenti and A. Consortini, "Estimate of characteristic scale of atmospheric turbulence by thin beams: comparison between the von Karman and Hill-Andrez models," J. Mod. Opt. 51, 333-342 (2003).
[CrossRef]

A. Consortini, Y. Y. Sun, C. Innocenti, and Z. P. Li, "Measuring inner scale of atmospheric turbulence by angle of arrival and scintillation," Opt. Commun. 216, 19-23 (2003).
[CrossRef]

N. Scafetta, L. Griffin, and B. J. West, "Hölder exponent spectra for human gait," Physica A 328, 561-583 (2003).
[CrossRef]

2002

R. B. Govindan, D. Vyushin, A. Bunde, S. Brenner, S. Havlin, and H. J. Schellnhuber, "Global climate models violate scaling of the observed atmospheric variability," Phys. Rev. Lett. 89, 0285011-0285014 (2002).
[CrossRef]

J. W. Kantelhardt, S. A. Zschiegner, E. Koscielny-Bunde, A. Bunde, S. Havlin, and H. E. Stanley, "Multifractal detrended fluctuations analysis of nonstationary time series," Physica A 316, 87-114 (2002).
[CrossRef]

2001

P. Ch. Ivanov, L. A. Nunes Amaral, S. Havlin, M. G. Rosenblum, H. E. Stanley, and Z. R. Struzik, "From 1/f noise to multifractal cascades in heartbeat dynamics," Chaos 11, 641-652 (2001).
[CrossRef]

S. Lovejoy, D. Schertzer, and J. D. Stanway, "Direct evidence of multifractal atmospheric cascades from planetary scales down to 1 km," Phys. Rev. Lett. 86, 5200-5204 (2001).
[CrossRef] [PubMed]

L. A. N. Amaral, P. Ch. Ivanov, N. Aoyagi, I. Hidaka, S. Tomono, A. L. Goldberger, H. E. Stanley, and Y. Yamamoto, "Behavioral-independent features of complex heartbeat dynamics," Phys. Rev. Lett. 86, 6026-6029 (2001).
[CrossRef]

1999

Y. Liu, P. Gopikrishnan, P. Cizeau, M. Meyer, C.-K. Peng, and H. E. Stanley, "Statistical properties of the volatility of price fluctuations," Phys. Rev. E 60, 1390-1400 (1999).
[CrossRef]

H. Okayama and L. Z. Wang, "Measurement of the spatial coherence of light influenced by turbulence," Appl. Opt. 38, 2342-2345 (1999).
[CrossRef]

1998

E. Koscielny-Bunde, A. Bunde, S. Havlin, H. E. Roman, Y. Goldreich, and H.-J. Schellnhuber, "Indication of a universal persistence law governing atmospheric variability," Phys. Rev. Lett. 81, 729-732 (1998).
[CrossRef]

1996

1994

L. C. Andrews, W. B. Miller, and J. C. Ricklin, "Spatial coherence of a Gaussian-beam wave in weak and strong optical turbulence," J. Opt. Soc. Am. A 11, 1653-1660 (1994).
[CrossRef]

F. Schmitt, D. Schertzer, S. Lovejoy, and Y. Brunet, "Empirical study of multifractal phase transitions in atmospheric turbulence," Nonlinear Processes Geophys. 1, 95-100 (1994).
[CrossRef]

C. K. Peng, S. V. Buldyrev, S. Havlin, M. Simons, H. E. Stanley, and A. L. Goldberger, "Mosaic organinazation of DNA nucleotides," Phys. Rev. E. 49, 1685-1689 (1994).
[CrossRef]

1989

R. F. Voss, "Random fractals self-affinity in noise, music mountains, and clouds," Physica D 38, 362-371 (1989).
[CrossRef]

1980

M. S. Belenkii and V. L. Mironov, "Coherence of the field of a laser beam in a turbulent atmosphere," Sov. J. Quantum Electron. 10, 595-597 (1980).
[CrossRef]

1967

Z. I. Feizulin and Yu. A. Kravtsov, "Broadening of a laser beam in a turbulent medium," Radio Quantum Electron. 10, 33-35 (1967).
[CrossRef]

Amaral, L. A. N.

L. A. N. Amaral, P. Ch. Ivanov, N. Aoyagi, I. Hidaka, S. Tomono, A. L. Goldberger, H. E. Stanley, and Y. Yamamoto, "Behavioral-independent features of complex heartbeat dynamics," Phys. Rev. Lett. 86, 6026-6029 (2001).
[CrossRef]

Amaral, L. A. Nunes

P. Ch. Ivanov, L. A. Nunes Amaral, S. Havlin, M. G. Rosenblum, H. E. Stanley, and Z. R. Struzik, "From 1/f noise to multifractal cascades in heartbeat dynamics," Chaos 11, 641-652 (2001).
[CrossRef]

Andrews, L. C.

Aoyagi, N.

L. A. N. Amaral, P. Ch. Ivanov, N. Aoyagi, I. Hidaka, S. Tomono, A. L. Goldberger, H. E. Stanley, and Y. Yamamoto, "Behavioral-independent features of complex heartbeat dynamics," Phys. Rev. Lett. 86, 6026-6029 (2001).
[CrossRef]

Belenkii, M. S.

M. S. Belenkii and V. L. Mironov, "Coherence of the field of a laser beam in a turbulent atmosphere," Sov. J. Quantum Electron. 10, 595-597 (1980).
[CrossRef]

Bowersox, R. D.

Braun, P.

E. Koscielny-Bunde, J. W. Kantelhardt, P. Braun, A. Bunde, and S. Havlin, "Long-term persistence and multifractality of river runoff records: detrended fluctuation studies," arXiv.org e-Print archive, physics/0305078, 19 May 2003, http://pbparxiv.org/abs/physics/0305078.

Brenner, S.

R. B. Govindan, D. Vyushin, A. Bunde, S. Brenner, S. Havlin, and H. J. Schellnhuber, "Global climate models violate scaling of the observed atmospheric variability," Phys. Rev. Lett. 89, 0285011-0285014 (2002).
[CrossRef]

Brunet, Y.

F. Schmitt, D. Schertzer, S. Lovejoy, and Y. Brunet, "Empirical study of multifractal phase transitions in atmospheric turbulence," Nonlinear Processes Geophys. 1, 95-100 (1994).
[CrossRef]

Buldyrev, S. V.

C. K. Peng, S. V. Buldyrev, S. Havlin, M. Simons, H. E. Stanley, and A. L. Goldberger, "Mosaic organinazation of DNA nucleotides," Phys. Rev. E. 49, 1685-1689 (1994).
[CrossRef]

Bunde, A.

J. W. Kantelhardt, S. A. Zschiegner, E. Koscielny-Bunde, A. Bunde, S. Havlin, and H. E. Stanley, "Multifractal detrended fluctuations analysis of nonstationary time series," Physica A 316, 87-114 (2002).
[CrossRef]

R. B. Govindan, D. Vyushin, A. Bunde, S. Brenner, S. Havlin, and H. J. Schellnhuber, "Global climate models violate scaling of the observed atmospheric variability," Phys. Rev. Lett. 89, 0285011-0285014 (2002).
[CrossRef]

E. Koscielny-Bunde, A. Bunde, S. Havlin, H. E. Roman, Y. Goldreich, and H.-J. Schellnhuber, "Indication of a universal persistence law governing atmospheric variability," Phys. Rev. Lett. 81, 729-732 (1998).
[CrossRef]

E. Koscielny-Bunde, J. W. Kantelhardt, P. Braun, A. Bunde, and S. Havlin, "Long-term persistence and multifractality of river runoff records: detrended fluctuation studies," arXiv.org e-Print archive, physics/0305078, 19 May 2003, http://pbparxiv.org/abs/physics/0305078.

Consortini, A.

C. Innocenti and A. Consortini, "Estimate of characteristic scale of atmospheric turbulence by thin beams: comparison between the von Karman and Hill-Andrez models," J. Mod. Opt. 51, 333-342 (2003).
[CrossRef]

A. Consortini, Y. Y. Sun, C. Innocenti, and Z. P. Li, "Measuring inner scale of atmospheric turbulence by angle of arrival and scintillation," Opt. Commun. 216, 19-23 (2003).
[CrossRef]

Feizulin, Z. I.

Z. I. Feizulin and Yu. A. Kravtsov, "Broadening of a laser beam in a turbulent medium," Radio Quantum Electron. 10, 33-35 (1967).
[CrossRef]

Gardner, P. J.

Goldberger, A. L.

L. A. N. Amaral, P. Ch. Ivanov, N. Aoyagi, I. Hidaka, S. Tomono, A. L. Goldberger, H. E. Stanley, and Y. Yamamoto, "Behavioral-independent features of complex heartbeat dynamics," Phys. Rev. Lett. 86, 6026-6029 (2001).
[CrossRef]

C. K. Peng, S. V. Buldyrev, S. Havlin, M. Simons, H. E. Stanley, and A. L. Goldberger, "Mosaic organinazation of DNA nucleotides," Phys. Rev. E. 49, 1685-1689 (1994).
[CrossRef]

Goldreich, Y.

E. Koscielny-Bunde, A. Bunde, S. Havlin, H. E. Roman, Y. Goldreich, and H.-J. Schellnhuber, "Indication of a universal persistence law governing atmospheric variability," Phys. Rev. Lett. 81, 729-732 (1998).
[CrossRef]

Gopikrishnan, P.

Y. Liu, P. Gopikrishnan, P. Cizeau, M. Meyer, C.-K. Peng, and H. E. Stanley, "Statistical properties of the volatility of price fluctuations," Phys. Rev. E 60, 1390-1400 (1999).
[CrossRef]

Govindan, R. B.

R. B. Govindan, D. Vyushin, A. Bunde, S. Brenner, S. Havlin, and H. J. Schellnhuber, "Global climate models violate scaling of the observed atmospheric variability," Phys. Rev. Lett. 89, 0285011-0285014 (2002).
[CrossRef]

Griffin, L.

N. Scafetta, L. Griffin, and B. J. West, "Hölder exponent spectra for human gait," Physica A 328, 561-583 (2003).
[CrossRef]

Havlin, S.

R. B. Govindan, D. Vyushin, A. Bunde, S. Brenner, S. Havlin, and H. J. Schellnhuber, "Global climate models violate scaling of the observed atmospheric variability," Phys. Rev. Lett. 89, 0285011-0285014 (2002).
[CrossRef]

J. W. Kantelhardt, S. A. Zschiegner, E. Koscielny-Bunde, A. Bunde, S. Havlin, and H. E. Stanley, "Multifractal detrended fluctuations analysis of nonstationary time series," Physica A 316, 87-114 (2002).
[CrossRef]

P. Ch. Ivanov, L. A. Nunes Amaral, S. Havlin, M. G. Rosenblum, H. E. Stanley, and Z. R. Struzik, "From 1/f noise to multifractal cascades in heartbeat dynamics," Chaos 11, 641-652 (2001).
[CrossRef]

E. Koscielny-Bunde, A. Bunde, S. Havlin, H. E. Roman, Y. Goldreich, and H.-J. Schellnhuber, "Indication of a universal persistence law governing atmospheric variability," Phys. Rev. Lett. 81, 729-732 (1998).
[CrossRef]

C. K. Peng, S. V. Buldyrev, S. Havlin, M. Simons, H. E. Stanley, and A. L. Goldberger, "Mosaic organinazation of DNA nucleotides," Phys. Rev. E. 49, 1685-1689 (1994).
[CrossRef]

E. Koscielny-Bunde, J. W. Kantelhardt, P. Braun, A. Bunde, and S. Havlin, "Long-term persistence and multifractality of river runoff records: detrended fluctuation studies," arXiv.org e-Print archive, physics/0305078, 19 May 2003, http://pbparxiv.org/abs/physics/0305078.

Hidaka, I.

L. A. N. Amaral, P. Ch. Ivanov, N. Aoyagi, I. Hidaka, S. Tomono, A. L. Goldberger, H. E. Stanley, and Y. Yamamoto, "Behavioral-independent features of complex heartbeat dynamics," Phys. Rev. Lett. 86, 6026-6029 (2001).
[CrossRef]

Innocenti, C.

A. Consortini, Y. Y. Sun, C. Innocenti, and Z. P. Li, "Measuring inner scale of atmospheric turbulence by angle of arrival and scintillation," Opt. Commun. 216, 19-23 (2003).
[CrossRef]

C. Innocenti and A. Consortini, "Estimate of characteristic scale of atmospheric turbulence by thin beams: comparison between the von Karman and Hill-Andrez models," J. Mod. Opt. 51, 333-342 (2003).
[CrossRef]

Ivanov, P. Ch.

L. A. N. Amaral, P. Ch. Ivanov, N. Aoyagi, I. Hidaka, S. Tomono, A. L. Goldberger, H. E. Stanley, and Y. Yamamoto, "Behavioral-independent features of complex heartbeat dynamics," Phys. Rev. Lett. 86, 6026-6029 (2001).
[CrossRef]

P. Ch. Ivanov, L. A. Nunes Amaral, S. Havlin, M. G. Rosenblum, H. E. Stanley, and Z. R. Struzik, "From 1/f noise to multifractal cascades in heartbeat dynamics," Chaos 11, 641-652 (2001).
[CrossRef]

Kantelhardt, J. W.

J. W. Kantelhardt, S. A. Zschiegner, E. Koscielny-Bunde, A. Bunde, S. Havlin, and H. E. Stanley, "Multifractal detrended fluctuations analysis of nonstationary time series," Physica A 316, 87-114 (2002).
[CrossRef]

E. Koscielny-Bunde, J. W. Kantelhardt, P. Braun, A. Bunde, and S. Havlin, "Long-term persistence and multifractality of river runoff records: detrended fluctuation studies," arXiv.org e-Print archive, physics/0305078, 19 May 2003, http://pbparxiv.org/abs/physics/0305078.

Koscielny-Bunde, E.

J. W. Kantelhardt, S. A. Zschiegner, E. Koscielny-Bunde, A. Bunde, S. Havlin, and H. E. Stanley, "Multifractal detrended fluctuations analysis of nonstationary time series," Physica A 316, 87-114 (2002).
[CrossRef]

E. Koscielny-Bunde, A. Bunde, S. Havlin, H. E. Roman, Y. Goldreich, and H.-J. Schellnhuber, "Indication of a universal persistence law governing atmospheric variability," Phys. Rev. Lett. 81, 729-732 (1998).
[CrossRef]

E. Koscielny-Bunde, J. W. Kantelhardt, P. Braun, A. Bunde, and S. Havlin, "Long-term persistence and multifractality of river runoff records: detrended fluctuation studies," arXiv.org e-Print archive, physics/0305078, 19 May 2003, http://pbparxiv.org/abs/physics/0305078.

Kravtsov, Yu. A.

Z. I. Feizulin and Yu. A. Kravtsov, "Broadening of a laser beam in a turbulent medium," Radio Quantum Electron. 10, 33-35 (1967).
[CrossRef]

Li, Z. P.

A. Consortini, Y. Y. Sun, C. Innocenti, and Z. P. Li, "Measuring inner scale of atmospheric turbulence by angle of arrival and scintillation," Opt. Commun. 216, 19-23 (2003).
[CrossRef]

Liu, Y.

Y. Liu, P. Gopikrishnan, P. Cizeau, M. Meyer, C.-K. Peng, and H. E. Stanley, "Statistical properties of the volatility of price fluctuations," Phys. Rev. E 60, 1390-1400 (1999).
[CrossRef]

Lovejoy, S.

S. Lovejoy, D. Schertzer, and J. D. Stanway, "Direct evidence of multifractal atmospheric cascades from planetary scales down to 1 km," Phys. Rev. Lett. 86, 5200-5204 (2001).
[CrossRef] [PubMed]

F. Schmitt, D. Schertzer, S. Lovejoy, and Y. Brunet, "Empirical study of multifractal phase transitions in atmospheric turbulence," Nonlinear Processes Geophys. 1, 95-100 (1994).
[CrossRef]

Luke, T. E.

Miller, W. B.

Mironov, V. L.

M. S. Belenkii and V. L. Mironov, "Coherence of the field of a laser beam in a turbulent atmosphere," Sov. J. Quantum Electron. 10, 595-597 (1980).
[CrossRef]

Okayama, H.

Peng, C. K.

C. K. Peng, S. V. Buldyrev, S. Havlin, M. Simons, H. E. Stanley, and A. L. Goldberger, "Mosaic organinazation of DNA nucleotides," Phys. Rev. E. 49, 1685-1689 (1994).
[CrossRef]

Pratt, W.

W. Pratt, Digital Image Processing (Wiley, 1991), pp. 630-634.

Ricklin, J. C.

Roggemann, M. C.

Roman, H. E.

E. Koscielny-Bunde, A. Bunde, S. Havlin, H. E. Roman, Y. Goldreich, and H.-J. Schellnhuber, "Indication of a universal persistence law governing atmospheric variability," Phys. Rev. Lett. 81, 729-732 (1998).
[CrossRef]

Rosenblum, M. G.

P. Ch. Ivanov, L. A. Nunes Amaral, S. Havlin, M. G. Rosenblum, H. E. Stanley, and Z. R. Struzik, "From 1/f noise to multifractal cascades in heartbeat dynamics," Chaos 11, 641-652 (2001).
[CrossRef]

Scafetta, N.

N. Scafetta, L. Griffin, and B. J. West, "Hölder exponent spectra for human gait," Physica A 328, 561-583 (2003).
[CrossRef]

Schellnhuber, H. J.

R. B. Govindan, D. Vyushin, A. Bunde, S. Brenner, S. Havlin, and H. J. Schellnhuber, "Global climate models violate scaling of the observed atmospheric variability," Phys. Rev. Lett. 89, 0285011-0285014 (2002).
[CrossRef]

Schellnhuber, H.-J.

E. Koscielny-Bunde, A. Bunde, S. Havlin, H. E. Roman, Y. Goldreich, and H.-J. Schellnhuber, "Indication of a universal persistence law governing atmospheric variability," Phys. Rev. Lett. 81, 729-732 (1998).
[CrossRef]

Schertzer, D.

S. Lovejoy, D. Schertzer, and J. D. Stanway, "Direct evidence of multifractal atmospheric cascades from planetary scales down to 1 km," Phys. Rev. Lett. 86, 5200-5204 (2001).
[CrossRef] [PubMed]

F. Schmitt, D. Schertzer, S. Lovejoy, and Y. Brunet, "Empirical study of multifractal phase transitions in atmospheric turbulence," Nonlinear Processes Geophys. 1, 95-100 (1994).
[CrossRef]

Schmitt, F.

F. Schmitt, D. Schertzer, S. Lovejoy, and Y. Brunet, "Empirical study of multifractal phase transitions in atmospheric turbulence," Nonlinear Processes Geophys. 1, 95-100 (1994).
[CrossRef]

Simons, M.

C. K. Peng, S. V. Buldyrev, S. Havlin, M. Simons, H. E. Stanley, and A. L. Goldberger, "Mosaic organinazation of DNA nucleotides," Phys. Rev. E. 49, 1685-1689 (1994).
[CrossRef]

Stanley, H. E.

J. W. Kantelhardt, S. A. Zschiegner, E. Koscielny-Bunde, A. Bunde, S. Havlin, and H. E. Stanley, "Multifractal detrended fluctuations analysis of nonstationary time series," Physica A 316, 87-114 (2002).
[CrossRef]

P. Ch. Ivanov, L. A. Nunes Amaral, S. Havlin, M. G. Rosenblum, H. E. Stanley, and Z. R. Struzik, "From 1/f noise to multifractal cascades in heartbeat dynamics," Chaos 11, 641-652 (2001).
[CrossRef]

L. A. N. Amaral, P. Ch. Ivanov, N. Aoyagi, I. Hidaka, S. Tomono, A. L. Goldberger, H. E. Stanley, and Y. Yamamoto, "Behavioral-independent features of complex heartbeat dynamics," Phys. Rev. Lett. 86, 6026-6029 (2001).
[CrossRef]

Y. Liu, P. Gopikrishnan, P. Cizeau, M. Meyer, C.-K. Peng, and H. E. Stanley, "Statistical properties of the volatility of price fluctuations," Phys. Rev. E 60, 1390-1400 (1999).
[CrossRef]

C. K. Peng, S. V. Buldyrev, S. Havlin, M. Simons, H. E. Stanley, and A. L. Goldberger, "Mosaic organinazation of DNA nucleotides," Phys. Rev. E. 49, 1685-1689 (1994).
[CrossRef]

Stanway, J. D.

S. Lovejoy, D. Schertzer, and J. D. Stanway, "Direct evidence of multifractal atmospheric cascades from planetary scales down to 1 km," Phys. Rev. Lett. 86, 5200-5204 (2001).
[CrossRef] [PubMed]

Struzik, Z. R.

P. Ch. Ivanov, L. A. Nunes Amaral, S. Havlin, M. G. Rosenblum, H. E. Stanley, and Z. R. Struzik, "From 1/f noise to multifractal cascades in heartbeat dynamics," Chaos 11, 641-652 (2001).
[CrossRef]

Sun, Y. Y.

A. Consortini, Y. Y. Sun, C. Innocenti, and Z. P. Li, "Measuring inner scale of atmospheric turbulence by angle of arrival and scintillation," Opt. Commun. 216, 19-23 (2003).
[CrossRef]

Tomono, S.

L. A. N. Amaral, P. Ch. Ivanov, N. Aoyagi, I. Hidaka, S. Tomono, A. L. Goldberger, H. E. Stanley, and Y. Yamamoto, "Behavioral-independent features of complex heartbeat dynamics," Phys. Rev. Lett. 86, 6026-6029 (2001).
[CrossRef]

Voss, R. F.

R. F. Voss, "Random fractals self-affinity in noise, music mountains, and clouds," Physica D 38, 362-371 (1989).
[CrossRef]

Vyushin, D.

R. B. Govindan, D. Vyushin, A. Bunde, S. Brenner, S. Havlin, and H. J. Schellnhuber, "Global climate models violate scaling of the observed atmospheric variability," Phys. Rev. Lett. 89, 0285011-0285014 (2002).
[CrossRef]

Wang, L. Z.

Welsh, B. M.

West, B. J.

N. Scafetta, L. Griffin, and B. J. West, "Hölder exponent spectra for human gait," Physica A 328, 561-583 (2003).
[CrossRef]

Yamamoto, Y.

L. A. N. Amaral, P. Ch. Ivanov, N. Aoyagi, I. Hidaka, S. Tomono, A. L. Goldberger, H. E. Stanley, and Y. Yamamoto, "Behavioral-independent features of complex heartbeat dynamics," Phys. Rev. Lett. 86, 6026-6029 (2001).
[CrossRef]

Zschiegner, S. A.

J. W. Kantelhardt, S. A. Zschiegner, E. Koscielny-Bunde, A. Bunde, S. Havlin, and H. E. Stanley, "Multifractal detrended fluctuations analysis of nonstationary time series," Physica A 316, 87-114 (2002).
[CrossRef]

Appl. Opt.

Chaos

P. Ch. Ivanov, L. A. Nunes Amaral, S. Havlin, M. G. Rosenblum, H. E. Stanley, and Z. R. Struzik, "From 1/f noise to multifractal cascades in heartbeat dynamics," Chaos 11, 641-652 (2001).
[CrossRef]

J. Mod. Opt.

C. Innocenti and A. Consortini, "Estimate of characteristic scale of atmospheric turbulence by thin beams: comparison between the von Karman and Hill-Andrez models," J. Mod. Opt. 51, 333-342 (2003).
[CrossRef]

J. Opt. Soc. Am. A

Nonlinear Processes Geophys.

F. Schmitt, D. Schertzer, S. Lovejoy, and Y. Brunet, "Empirical study of multifractal phase transitions in atmospheric turbulence," Nonlinear Processes Geophys. 1, 95-100 (1994).
[CrossRef]

Opt. Commun.

A. Consortini, Y. Y. Sun, C. Innocenti, and Z. P. Li, "Measuring inner scale of atmospheric turbulence by angle of arrival and scintillation," Opt. Commun. 216, 19-23 (2003).
[CrossRef]

Phys. Rev. E

Y. Liu, P. Gopikrishnan, P. Cizeau, M. Meyer, C.-K. Peng, and H. E. Stanley, "Statistical properties of the volatility of price fluctuations," Phys. Rev. E 60, 1390-1400 (1999).
[CrossRef]

Phys. Rev. E.

C. K. Peng, S. V. Buldyrev, S. Havlin, M. Simons, H. E. Stanley, and A. L. Goldberger, "Mosaic organinazation of DNA nucleotides," Phys. Rev. E. 49, 1685-1689 (1994).
[CrossRef]

Phys. Rev. Lett.

E. Koscielny-Bunde, A. Bunde, S. Havlin, H. E. Roman, Y. Goldreich, and H.-J. Schellnhuber, "Indication of a universal persistence law governing atmospheric variability," Phys. Rev. Lett. 81, 729-732 (1998).
[CrossRef]

R. B. Govindan, D. Vyushin, A. Bunde, S. Brenner, S. Havlin, and H. J. Schellnhuber, "Global climate models violate scaling of the observed atmospheric variability," Phys. Rev. Lett. 89, 0285011-0285014 (2002).
[CrossRef]

S. Lovejoy, D. Schertzer, and J. D. Stanway, "Direct evidence of multifractal atmospheric cascades from planetary scales down to 1 km," Phys. Rev. Lett. 86, 5200-5204 (2001).
[CrossRef] [PubMed]

L. A. N. Amaral, P. Ch. Ivanov, N. Aoyagi, I. Hidaka, S. Tomono, A. L. Goldberger, H. E. Stanley, and Y. Yamamoto, "Behavioral-independent features of complex heartbeat dynamics," Phys. Rev. Lett. 86, 6026-6029 (2001).
[CrossRef]

Physica A

N. Scafetta, L. Griffin, and B. J. West, "Hölder exponent spectra for human gait," Physica A 328, 561-583 (2003).
[CrossRef]

J. W. Kantelhardt, S. A. Zschiegner, E. Koscielny-Bunde, A. Bunde, S. Havlin, and H. E. Stanley, "Multifractal detrended fluctuations analysis of nonstationary time series," Physica A 316, 87-114 (2002).
[CrossRef]

Physica D

R. F. Voss, "Random fractals self-affinity in noise, music mountains, and clouds," Physica D 38, 362-371 (1989).
[CrossRef]

Radio Quantum Electron.

Z. I. Feizulin and Yu. A. Kravtsov, "Broadening of a laser beam in a turbulent medium," Radio Quantum Electron. 10, 33-35 (1967).
[CrossRef]

Sov. J. Quantum Electron.

M. S. Belenkii and V. L. Mironov, "Coherence of the field of a laser beam in a turbulent atmosphere," Sov. J. Quantum Electron. 10, 595-597 (1980).
[CrossRef]

Other

W. Pratt, Digital Image Processing (Wiley, 1991), pp. 630-634.

E. Koscielny-Bunde, J. W. Kantelhardt, P. Braun, A. Bunde, and S. Havlin, "Long-term persistence and multifractality of river runoff records: detrended fluctuation studies," arXiv.org e-Print archive, physics/0305078, 19 May 2003, http://pbparxiv.org/abs/physics/0305078.

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Figures (12)

Fig. 1
Fig. 1

Setup used for the measurement of fractal properties of a beam passing through a turbulent medium of length L = 1.2 m. The angle between the two opposite paths in the turbulent medium is about 15°. The transverse dimension LT of the turbulent medium is about 30 cm. L1 and L2 are the beam expander lenses.

Fig. 2
Fig. 2

(a) Beam diameter at the half-maximum intensity as a function of the wind velocity. The beam diameter measured is normalized to the reference beam without turbulence. The fit is done with a power law d = aVb with a = 0.88 and b = 0.1. (b) Beam surface at the half-maximum intensity as a function of the wind velocity. The curve is fitted with a power law S = aVb with a = 0.91 and b = 0.04. A value of 100% corresponds to no change with the reference beam.

Fig. 3
Fig. 3

Laser beam wandering as a function of the wind velocity.

Fig. 4
Fig. 4

(Color online) Fringe visibility as a function of the interslit distance b and with a slit's width of a = 420 μm and for two wind velocities of 8.8 and 4.1 m∕s.

Fig. 5
Fig. 5

(Color online) q dependence of τ(q) with τ(q) = qh(q) − 1.

Fig. 6
Fig. 6

Generalized Hurst exponent h(q) calculated for q = 2 and given by the slope of F 2(s) as a function of the wind velocity. For q = 2, the standard DFA is retrieved. The different symbols correspond to different experimental measurements done at different moments, whereas the curve corresponds to the mean value.

Fig. 7
Fig. 7

(Color online) q dependence of the generalized Hurst exponent h(q) given by the slope of Fq (s) by analyzing log–log plots Fq (s). Fq (s) ≈ sh(q ) .

Fig. 8
Fig. 8

(Color online) Comparison between experimental values of h(q) given by the MF-DFA algorithm and the fit for two values of wind velocities obtained with Eq. (13).

Fig. 9
Fig. 9

(Color online) Singularity spectrum f(α) determined by the modified Legendre transform. α is the singularity strength, or Hölder exponent, whereas f(α) denotes the dimension of the subset of the laser beam intensity that is characterized by α. f(α) = q[α − h(q)] + 1. Also plotted are experimental values of f(α) given by the MF-DFA algorithm and the fit for different values of wind velocities.

Fig. 10
Fig. 10

Width of the singularity spectrum f(α) at f(α = 0) versus the wind velocity. β is the slope of the curve.

Fig. 11
Fig. 11

Images of intensity patterns of the laser beam for different wind velocities.

Fig. 12
Fig. 12

Color-coded wavelet analysis of a transverse spatial intensity of a laser beam superposed with the intensity cross section for the reference beam propagating without turbulence and the beam after propagation in a medium with different wind velocities. The x axis represents the transverse coordinate of the laser intensity pattern and the y axis indicates the scale of the wavelet used (Mexican hat) in normalized frequency, with the large scale at the top. A horizontal cut through the figure gives the fluctuations on a specific spatial scale. The brighter colors indicate larger values of the wavelet amplitudes corresponding to large intensity fluctuations.

Tables (1)

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Table 1 Parameters for Fitting the Exponent Range h ( q )

Equations (13)

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e z = L T V ν ,
S   or   d = a V b ,
V = I max I min I max I min = 2 ( I 1 I 2 ) 1 / 2 I 1 + I 2 | γ 12 | ,
γ exp { 1.455 C n     2 k 2 L r 12 5 / 3 [ 1 ( 5 6 L 2 k 2 a 4 ) ] } ,
F 2 ( s , ν ) = 1 s i = 1 s { | Y [ ( ν 1 ) s + i ] y ν ( i ) | } 2 ,
F q ( s ) = { 1 N s ν = 1 N s [ F 2 ( ν , s ) ] q / 2 } 1 / q .
F q ( s ) s h ( q ) ,
h ( q ) = [ τ ( q ) + 1 ] q
α = d τ ( q ) d q
α = h ( q ) + q d h ( q ) d q ,
f ( α ) = q [ α h ( q ) ] + 1.
h ( q ) = 1 q ln ( a q + b q ) q ln 2 .
h q = α q + β 1 q ln ( a q + b q ) q ln 2 .

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