Abstract

We have previously shown that azopolymer thin films exposed to coherent light that has travelled through a turbulent medium produces a surface relief grating containing information about the intensity of the turbulence; for instance, a relation between the refractive index structure constant Cn2 as a function of the surface parameters was obtained. In this work, we show that these films capture much more information about the turbulence dynamics. Multifractal detrended fluctuation and fractal dimension analysis from images of the surface roughness produced by the light on the azopolymer reveals scaling properties related to those of the optical turbulence.

© 2014 Optical Society of America

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References

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  1. R. G. Buser, “Interferometric determination of the distance dependence of the phase structure function for near-ground horizontal propagation at 6328 Å,” J. Opt. Soc. Am. 61, 488–491 (1971).
    [CrossRef]
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  3. A. Berdja, “On the DIMM interpretation of non-Kolmogorov turbulence in the atmospheric surface layer,” Mon. Not. R. Astron. Soc. 409, 722–726 (2010).
    [CrossRef]
  4. D. G. Pérez and L. Zunino, “Generalized wave-front phase for non-Kolmogorov turbulence,” Opt. Lett. 33, 572–574 (2008).
    [CrossRef]
  5. F. Schmitt, D. Schertzer, S. Lovejoy, and Y. Brunet, “Multifractal temperature and fux of temperature variance in fully developed turbulence,” Europhys. Lett. 34, 195–200 (1996).
    [CrossRef]
  6. R. R. Prasad and K. R. Sreenivasan, “Quantitative three-dimensional imaging and the structure of passive scalar fields in fully turbulent flows,” J. Fluid Mech. 216, 1–34 (2006).
    [CrossRef]
  7. R. Rao, “Statistics of the fractal structure and phase singularity of a plane light wave propagation in atmospheric turbulence,” Appl. Opt. 47, 269–276 (2008).
    [CrossRef] [PubMed]
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    [CrossRef]
  9. J. W. Kantelhardt, S. A. Zschiegner, E. Koscielny-Bunde, A. Bunde, S. Havlin, and H. E. Stanley, “Multifractal detrended fluctuation analysis of nonstationary time series,” Physica A 316, 87–114 (2002).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  14. 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]
  15. K. J. Falconer, Fractal Geometry: Mathematical Theory and Applications (John Wiley, 1990).
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    [CrossRef]
  17. R. Barille and P. LaPenna, “Multifractality of laser beam spatial intensity in a turbulent medium,” Appl. Opt. 45, 3331–3339 (2006).
    [CrossRef] [PubMed]
  18. L. Zunino, M. C. Soriano, A. Figliola, D. G. Pérez, M. Garavaglia, C. R. Mirasso, and O. A. Rosso, “Performance of encryption schemes in chaotic optical communication: A multifractal approach,” Opt. Commun. 282, 4587–4594 (2009).
    [CrossRef]
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    [CrossRef] [PubMed]
  20. A. Arneodo, E. Bacry, and J. Muzy, “The thermodynamics of fractals revisited with wavelets,” Physica A 213, 232–275 (1995).
    [CrossRef]
  21. To change from iso-curves fractal dimension to graph dimension a 1 should be added to the former to obtain the latter [15].

2013 (2)

G. Funes, E. Figueroa, D. Gulich, L. Zunino, and D. G. Pérez, “Characterizing inertial and convective optical turbulence by detrended fluctuation analysis,” Proc. SPIE 8890, 889016 (2013).
[CrossRef]

R. Barillé, D. G. Pérez, Y. Morille, S. Zielińska, and E. Ortyl, “Simple turbulence measurements with azopolymer thin films,” Opt. Lett. 38, 1128–1130 (2013).
[CrossRef] [PubMed]

2012 (1)

E. A. F. Ihlen, “Introduction to multifractal detrended fluctuation analysis in matlab,” Front. Physiol. 3, 141 (2012).
[CrossRef] [PubMed]

2010 (1)

A. Berdja, “On the DIMM interpretation of non-Kolmogorov turbulence in the atmospheric surface layer,” Mon. Not. R. Astron. Soc. 409, 722–726 (2010).
[CrossRef]

2009 (1)

L. Zunino, M. C. Soriano, A. Figliola, D. G. Pérez, M. Garavaglia, C. R. Mirasso, and O. A. Rosso, “Performance of encryption schemes in chaotic optical communication: A multifractal approach,” Opt. Commun. 282, 4587–4594 (2009).
[CrossRef]

2008 (3)

2006 (3)

R. R. Prasad and K. R. Sreenivasan, “Quantitative three-dimensional imaging and the structure of passive scalar fields in fully turbulent flows,” J. Fluid Mech. 216, 1–34 (2006).
[CrossRef]

G.-F. Gu and W.-X. Zhou, “Detrended fluctuation analysis for fractals and multifractals in higher dimensions,” Phys. Rev. E 74, 061104 (2006).
[CrossRef]

R. Barille and P. LaPenna, “Multifractality of laser beam spatial intensity in a turbulent medium,” Appl. Opt. 45, 3331–3339 (2006).
[CrossRef] [PubMed]

2005 (1)

2003 (3)

E. Ortyl and S. Kucharski, “Refractive index modulation in polymeric photochromic films,” Cent. Eur. J. Chem. 1, 137–159 (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]

A. N. D. Posadas, D. Giménez, R. Quiroz, and R. Protz, “Multifractal characterization of soil pore systems,” Soil Sci. Soc. Am. J. 67, 1361–1369 (2003).
[CrossRef]

2002 (1)

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

1996 (1)

F. Schmitt, D. Schertzer, S. Lovejoy, and Y. Brunet, “Multifractal temperature and fux of temperature variance in fully developed turbulence,” Europhys. Lett. 34, 195–200 (1996).
[CrossRef]

1995 (1)

A. Arneodo, E. Bacry, and J. Muzy, “The thermodynamics of fractals revisited with wavelets,” Physica A 213, 232–275 (1995).
[CrossRef]

1971 (1)

Arneodo, A.

A. Arneodo, E. Bacry, and J. Muzy, “The thermodynamics of fractals revisited with wavelets,” Physica A 213, 232–275 (1995).
[CrossRef]

Bacry, E.

A. Arneodo, E. Bacry, and J. Muzy, “The thermodynamics of fractals revisited with wavelets,” Physica A 213, 232–275 (1995).
[CrossRef]

Barille, R.

Barillé, R.

Berdja, A.

A. Berdja, “On the DIMM interpretation of non-Kolmogorov turbulence in the atmospheric surface layer,” Mon. Not. R. Astron. Soc. 409, 722–726 (2010).
[CrossRef]

Brunet, Y.

F. Schmitt, D. Schertzer, S. Lovejoy, and Y. Brunet, “Multifractal temperature and fux of temperature variance in fully developed turbulence,” Europhys. Lett. 34, 195–200 (1996).
[CrossRef]

Bunde, A.

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

Buser, R. G.

Consortini, A.

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]

Dabos-Seignon, S.

Du, W.

Falconer, K. J.

K. J. Falconer, Fractal Geometry: Mathematical Theory and Applications (John Wiley, 1990).

Figliola, A.

L. Zunino, M. C. Soriano, A. Figliola, D. G. Pérez, M. Garavaglia, C. R. Mirasso, and O. A. Rosso, “Performance of encryption schemes in chaotic optical communication: A multifractal approach,” Opt. Commun. 282, 4587–4594 (2009).
[CrossRef]

Figueroa, E.

G. Funes, E. Figueroa, D. Gulich, L. Zunino, and D. G. Pérez, “Characterizing inertial and convective optical turbulence by detrended fluctuation analysis,” Proc. SPIE 8890, 889016 (2013).
[CrossRef]

Funes, G.

G. Funes, E. Figueroa, D. Gulich, L. Zunino, and D. G. Pérez, “Characterizing inertial and convective optical turbulence by detrended fluctuation analysis,” Proc. SPIE 8890, 889016 (2013).
[CrossRef]

Garavaglia, M.

L. Zunino, M. C. Soriano, A. Figliola, D. G. Pérez, M. Garavaglia, C. R. Mirasso, and O. A. Rosso, “Performance of encryption schemes in chaotic optical communication: A multifractal approach,” Opt. Commun. 282, 4587–4594 (2009).
[CrossRef]

Giménez, D.

A. N. D. Posadas, D. Giménez, R. Quiroz, and R. Protz, “Multifractal characterization of soil pore systems,” Soil Sci. Soc. Am. J. 67, 1361–1369 (2003).
[CrossRef]

Gu, G.-F.

G.-F. Gu and W.-X. Zhou, “Detrended fluctuation analysis for fractals and multifractals in higher dimensions,” Phys. Rev. E 74, 061104 (2006).
[CrossRef]

Gulich, D.

G. Funes, E. Figueroa, D. Gulich, L. Zunino, and D. G. Pérez, “Characterizing inertial and convective optical turbulence by detrended fluctuation analysis,” Proc. SPIE 8890, 889016 (2013).
[CrossRef]

Havlin, S.

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

Ihlen, E. A. F.

E. A. F. Ihlen, “Introduction to multifractal detrended fluctuation analysis in matlab,” Front. Physiol. 3, 141 (2012).
[CrossRef] [PubMed]

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]

Jiang, Y.

Kandjani, S. A.

Kantelhardt, J. W.

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

Koscielny-Bunde, E.

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

Kucharski, S.

LaPenna, P.

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]

Lovejoy, S.

F. Schmitt, D. Schertzer, S. Lovejoy, and Y. Brunet, “Multifractal temperature and fux of temperature variance in fully developed turbulence,” Europhys. Lett. 34, 195–200 (1996).
[CrossRef]

Ma, J.

Mirasso, C. R.

L. Zunino, M. C. Soriano, A. Figliola, D. G. Pérez, M. Garavaglia, C. R. Mirasso, and O. A. Rosso, “Performance of encryption schemes in chaotic optical communication: A multifractal approach,” Opt. Commun. 282, 4587–4594 (2009).
[CrossRef]

Morille, Y.

Muzy, J.

A. Arneodo, E. Bacry, and J. Muzy, “The thermodynamics of fractals revisited with wavelets,” Physica A 213, 232–275 (1995).
[CrossRef]

Nunzi, J.-M.

Ortyl, E.

Pérez, D. G.

R. Barillé, D. G. Pérez, Y. Morille, S. Zielińska, and E. Ortyl, “Simple turbulence measurements with azopolymer thin films,” Opt. Lett. 38, 1128–1130 (2013).
[CrossRef] [PubMed]

G. Funes, E. Figueroa, D. Gulich, L. Zunino, and D. G. Pérez, “Characterizing inertial and convective optical turbulence by detrended fluctuation analysis,” Proc. SPIE 8890, 889016 (2013).
[CrossRef]

L. Zunino, M. C. Soriano, A. Figliola, D. G. Pérez, M. Garavaglia, C. R. Mirasso, and O. A. Rosso, “Performance of encryption schemes in chaotic optical communication: A multifractal approach,” Opt. Commun. 282, 4587–4594 (2009).
[CrossRef]

D. G. Pérez and L. Zunino, “Generalized wave-front phase for non-Kolmogorov turbulence,” Opt. Lett. 33, 572–574 (2008).
[CrossRef]

Posadas, A. N. D.

A. N. D. Posadas, D. Giménez, R. Quiroz, and R. Protz, “Multifractal characterization of soil pore systems,” Soil Sci. Soc. Am. J. 67, 1361–1369 (2003).
[CrossRef]

Prasad, R. R.

R. R. Prasad and K. R. Sreenivasan, “Quantitative three-dimensional imaging and the structure of passive scalar fields in fully turbulent flows,” J. Fluid Mech. 216, 1–34 (2006).
[CrossRef]

Protz, R.

A. N. D. Posadas, D. Giménez, R. Quiroz, and R. Protz, “Multifractal characterization of soil pore systems,” Soil Sci. Soc. Am. J. 67, 1361–1369 (2003).
[CrossRef]

Quiroz, R.

A. N. D. Posadas, D. Giménez, R. Quiroz, and R. Protz, “Multifractal characterization of soil pore systems,” Soil Sci. Soc. Am. J. 67, 1361–1369 (2003).
[CrossRef]

Rao, R.

Rosso, O. A.

L. Zunino, M. C. Soriano, A. Figliola, D. G. Pérez, M. Garavaglia, C. R. Mirasso, and O. A. Rosso, “Performance of encryption schemes in chaotic optical communication: A multifractal approach,” Opt. Commun. 282, 4587–4594 (2009).
[CrossRef]

Schertzer, D.

F. Schmitt, D. Schertzer, S. Lovejoy, and Y. Brunet, “Multifractal temperature and fux of temperature variance in fully developed turbulence,” Europhys. Lett. 34, 195–200 (1996).
[CrossRef]

Schmitt, F.

F. Schmitt, D. Schertzer, S. Lovejoy, and Y. Brunet, “Multifractal temperature and fux of temperature variance in fully developed turbulence,” Europhys. Lett. 34, 195–200 (1996).
[CrossRef]

Soriano, M. C.

L. Zunino, M. C. Soriano, A. Figliola, D. G. Pérez, M. Garavaglia, C. R. Mirasso, and O. A. Rosso, “Performance of encryption schemes in chaotic optical communication: A multifractal approach,” Opt. Commun. 282, 4587–4594 (2009).
[CrossRef]

Sreenivasan, K. R.

R. R. Prasad and K. R. Sreenivasan, “Quantitative three-dimensional imaging and the structure of passive scalar fields in fully turbulent flows,” J. Fluid Mech. 216, 1–34 (2006).
[CrossRef]

Stanley, H. E.

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

Tan, L.

Yu, S.

Zhou, W.-X.

G.-F. Gu and W.-X. Zhou, “Detrended fluctuation analysis for fractals and multifractals in higher dimensions,” Phys. Rev. E 74, 061104 (2006).
[CrossRef]

Zielinska, S.

Zschiegner, S. A.

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

Zunino, L.

G. Funes, E. Figueroa, D. Gulich, L. Zunino, and D. G. Pérez, “Characterizing inertial and convective optical turbulence by detrended fluctuation analysis,” Proc. SPIE 8890, 889016 (2013).
[CrossRef]

L. Zunino, M. C. Soriano, A. Figliola, D. G. Pérez, M. Garavaglia, C. R. Mirasso, and O. A. Rosso, “Performance of encryption schemes in chaotic optical communication: A multifractal approach,” Opt. Commun. 282, 4587–4594 (2009).
[CrossRef]

D. G. Pérez and L. Zunino, “Generalized wave-front phase for non-Kolmogorov turbulence,” Opt. Lett. 33, 572–574 (2008).
[CrossRef]

Appl. Opt. (2)

Cent. Eur. J. Chem. (1)

E. Ortyl and S. Kucharski, “Refractive index modulation in polymeric photochromic films,” Cent. Eur. J. Chem. 1, 137–159 (2003).
[CrossRef]

Europhys. Lett. (1)

F. Schmitt, D. Schertzer, S. Lovejoy, and Y. Brunet, “Multifractal temperature and fux of temperature variance in fully developed turbulence,” Europhys. Lett. 34, 195–200 (1996).
[CrossRef]

Front. Physiol. (1)

E. A. F. Ihlen, “Introduction to multifractal detrended fluctuation analysis in matlab,” Front. Physiol. 3, 141 (2012).
[CrossRef] [PubMed]

J. Fluid Mech. (1)

R. R. Prasad and K. R. Sreenivasan, “Quantitative three-dimensional imaging and the structure of passive scalar fields in fully turbulent flows,” J. Fluid Mech. 216, 1–34 (2006).
[CrossRef]

J. Opt. Soc. Am. (1)

Mon. Not. R. Astron. Soc. (1)

A. Berdja, “On the DIMM interpretation of non-Kolmogorov turbulence in the atmospheric surface layer,” Mon. Not. R. Astron. Soc. 409, 722–726 (2010).
[CrossRef]

Opt. Commun. (2)

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]

L. Zunino, M. C. Soriano, A. Figliola, D. G. Pérez, M. Garavaglia, C. R. Mirasso, and O. A. Rosso, “Performance of encryption schemes in chaotic optical communication: A multifractal approach,” Opt. Commun. 282, 4587–4594 (2009).
[CrossRef]

Opt. Express (1)

Opt. Lett. (3)

Phys. Rev. E (1)

G.-F. Gu and W.-X. Zhou, “Detrended fluctuation analysis for fractals and multifractals in higher dimensions,” Phys. Rev. E 74, 061104 (2006).
[CrossRef]

Physica A (2)

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

A. Arneodo, E. Bacry, and J. Muzy, “The thermodynamics of fractals revisited with wavelets,” Physica A 213, 232–275 (1995).
[CrossRef]

Proc. SPIE (1)

G. Funes, E. Figueroa, D. Gulich, L. Zunino, and D. G. Pérez, “Characterizing inertial and convective optical turbulence by detrended fluctuation analysis,” Proc. SPIE 8890, 889016 (2013).
[CrossRef]

Soil Sci. Soc. Am. J. (1)

A. N. D. Posadas, D. Giménez, R. Quiroz, and R. Protz, “Multifractal characterization of soil pore systems,” Soil Sci. Soc. Am. J. 67, 1361–1369 (2003).
[CrossRef]

Other (2)

To change from iso-curves fractal dimension to graph dimension a 1 should be added to the former to obtain the latter [15].

K. J. Falconer, Fractal Geometry: Mathematical Theory and Applications (John Wiley, 1990).

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Figures (5)

Fig. 1
Fig. 1

Experimental set-up to measure the influence of a turbid medium on the laser beam propagation. The heaters are able to modify the refractive index, generating turbulence.

Fig. 2
Fig. 2

Surface patterns on the materials obtained for four different turbulent conditions (left). The central patterns correspond to the contours of the Gaussian surfaces by thresholding the height of the topography with a fixed spacing Δ between the height of the successive contours (center). Filling of the central patterns corresponding to the contours of the gaussian surfaces (right). The four patterns corresponds to C n 2 = 9.4 × 10 13, 1.4 × 10−12, 3.8 × 10−12, and 4.1 × 10−12 m−2/3.

Fig. 3
Fig. 3

Box-counting dimension of the patterns as a function of the refractive index structure constant C n 2.

Fig. 4
Fig. 4

The Hurst (a) and multifractal scaling (b) exponents as functions of the qth moment for different turbulence intensities.

Fig. 5
Fig. 5

(a) Singlularity spectrum f (α) as a function of the Hölder exponent for different turbulences estimated from the thin-film patterns. (b) Maximum of the singularity spectrum as a function of the structure constant of the refractive index, C n 2.

Equations (4)

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

u v , w ( i , j ) = m , n = 1 i , j [ X v , w ( n , m ) X u , w ]
F 2 ( v , w , s ) = 1 s 2 i , j = 1 s , s ε v , w ( i , j ) 2 ;
F q ( s ) = { 1 M s N s v , w = 1 M s , N s [ F 2 ( v , w , s ) ] q / 2 } 1 / q .
α = d τ ( q ) d q = h ( q ) + q d h d q ( q ) and f ( α ) = q ( α ) α τ ( q ( α ) ) = q [ α h ( q ) ] + 2 .

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