Abstract

We introduce the Lévy fractional Brownian field family to model the turbulent wavefront phase. This generalized model allows us to overcome the limitations found in a previous approach [Perez et al., J. Opt. Soc. Am. A 21, 1962 (2004)]. More precisely, our new model provides stationary phase increments over the full inertial range. Thus it successfully extends classical results to non-Kolmogorov turbulence without any approximation.

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References

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2007 (1)

J.-M. Bardet and P. Bertrand, J. Time Ser. Anal. 28, 1 (2007).
[CrossRef]

2006 (1)

E. Herbin, Rocky Mountain J. Math. 36, 1249 (2006).
[CrossRef]

2004 (1)

2003 (1)

R. J. Elliott and J. van der Hoek, Math. Finance 13, 301 (2003).
[CrossRef]

2000 (1)

1998 (1)

1997 (2)

1996 (1)

1995 (2)

1994 (1)

1993 (1)

S. S. Olivier, C. E. Max, D. T. Gavel, and J. M. Brase, Astrophys. J. 407, 428 (1993).
[CrossRef]

1992 (2)

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

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

1990 (1)

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

1971 (1)

1966 (1)

Appl. Opt. (4)

Astron. Astrophys. (1)

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

Astrophys. J. (2)

S. S. Olivier, C. E. Max, D. T. Gavel, and J. M. Brase, Astrophys. J. 407, 428 (1993).
[CrossRef]

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

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (3)

J. Time Ser. Anal. (1)

J.-M. Bardet and P. Bertrand, J. Time Ser. Anal. 28, 1 (2007).
[CrossRef]

Math. Finance (1)

R. J. Elliott and J. van der Hoek, Math. Finance 13, 301 (2003).
[CrossRef]

Opt. Lett. (3)

Rocky Mountain J. Math. (1)

E. Herbin, Rocky Mountain J. Math. 36, 1249 (2006).
[CrossRef]

Other (2)

G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance, Stochastic Modeling (Chapman & Hall/CRC1994).

V. I. Tatarskĭ, Wave Propagation in a Turbulent Atmosphere (Nauka, 1967). [In Russian; English translation: The Effect of the Turbulent Atmosphere on Wave Propagation (NTIS, 1971)].

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Equations (12)

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D φ ( r r ) = C φ , β 2 ( r r r 0 , β ) β 2 ,
E [ B H ( r ) B H ( r ) ] = σ 2 2 ( r 2 H + r 2 H r r 2 H ) ;
B H ( r ) = 1 2 C 2 , H R 2 ( e i r ν 1 ) ν H + 1 W ̂ ( d 2 ν ) ,
φ ( R ρ ) = A φ , H R H B H ( ρ ) , ρ 1 ,
D φ [ R ( ρ ρ ) ] = A φ , H 2 R 2 H ρ ρ 2 H
A φ , H 2 = 2 2 H + 1 Γ H ( 1 + 1 H ) r 0 , H 2 H .
ϕ ( R ρ ) = A φ , H R H [ B H ( ρ ) 1 π ρ 1 B H ( ρ ) d 2 ρ ] .
A φ , H 2 R 2 H { 1 1 + H π C 2 , H 0 [ 2 J 1 ( ν ) ν 1 ] 2 ν 2 H + 1 d ν } = A φ , H 2 R 2 H Γ ( 2 + 2 H ) Γ ( 2 + H ) Γ ( 3 + H ) .
A φ , H 2 = 2 2 H Γ ( 2 + H ) Γ ( 3 + H ) Γ ( 2 + 2 H ) r 0 , H 2 H .
α ( r ) = λ 2 π φ ( r ) = C z , H r 0 , H 2 H B H ( r ) ,
σ m , H 2 ( ϕ r ) = E α ( ϕ r ) 2 = 2 π 2 C z , H 2 r 0 , H 2 H C 2 , H R 2 | ϕ ̂ ( ν ) | 2 ν 2 H d 2 ν ,
σ m , H 2 ( ϕ r ) = 4 H Γ ( H + 1 2 ) π 1 2 Γ ( 2 + H ) C φ , H 2 2 π 2 λ 2 D 2 H 2 r 0 , H 2 H .

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