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Generalized wavefront phase for non-Kolmogorov turbulence Darío G. Pérez1,* and Luciano Zunino2,3 1
Instituto de Física, Pontificia Universidad Católica de Valparaíso, 23-40025 Valparaíso, Chile 2 Centro de Investigaciones Ópticas, CC. 124 Correo Central, 1900 La Plata, Argentina 3 Departamento de Ciencias Básicas, Facultad de Ingeniería, Universidad Nacional de La Plata, 1900 La Plata, Argentina *Corresponding author:
[email protected] Received November 2, 2007; revised February 7, 2008; accepted February 10, 2008; posted February 13, 2008 (Doc. ID 89329); published March 11, 2008 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. © 2008 Optical Society of America OCIS codes: 000.5490, 010.1290, 010.7060, 010.7350.
The classical Obukhov–Kolmogorov (OK) theory had a fundamental role in the development of atmospheric and adaptive optics, but with the advent of high-angular-resolution techniques, e.g., stellar interferometry, departures from it have been observed. Several experiments have confirmed [1–5] that this model is sometimes incomplete to describe atmospheric statistics properly. In particular, for near-tothe-ground measurements, exponents in the range 共1 , 5 / 3兲 have been experimentally found. Any turbulent state that deviates from the 5 / 3-power law is classified as a non-Kolmogorov turbulence. The phase-structure function can be generalized [5] to include these observations: D共r⬘ − r兲 =
C2 ,
冉
储r⬘ − r储 r0,
冊
−2
,
共1兲
where  is the exponent associated with the phase spectrum, C2 , is a constant maintaining consistency between the power spectrum and the structure function of phase fluctuations, and r0, is the atmospheric coherence diameter but now is usually known as the generalized Fried parameter—for  = 11/ 3, the classical OK model is recovered. Under this definition the turbulent wavefront phase is a zero-mean process, and the structure function implies it has stationary increments. There is only one one-dimensional Gaussian process representative of these properties: the fractional Brownian motion (fBm) with self-similar index H, better known as the Hurst parameter, BH [6]. Schwartz et al. [7] were the first to propose fBm processes to model the turbulence-degraded wavefront phase in the OK case. They considered these wavefronts as (Gaussian) realizations of fractal surfaces with a fractal dimension equal to 13/ 6. Then, under the Taylor’s hypothesis, these surfaces were mapped to a fBm with Hurst parameter H = 5 / 6. But this assertion only had some relevance as a justification to 0146-9592/08/060572-3/$15.00
the use of predictive algorithms in atmospheric optics, temporal analysis of turbulent wavefront-tilt data, or numerical modeling of turbulent wavefronts [8–12]. In any case, most of them considered only the Kolmogorov case and were unrelated to the nonKolmogorov models discussed previously. Moreover, the turbulent wavefront is a twodimensional scalar field, but because of the lack of an analytical model these works relay in numerical models or time-series analysis. Recently, an analytical approach based on the fBm concept was given in Pérez et al. [13]. We presented a generalized version that allows H to take values other than 5 / 6, called an “isotropic” fractional Brownian motion model (ifBm); in this way, extending the proposal of Schwartz et al. to the non-Kolmogorov case. Also, it should be stressed that this model is not based on the spectral analysis; it cannot be used, without tweaking, because self-similar processes are not stationary—see [6], Chapter 7. Instead, we use a general fractional white noise analysis [14] to express the wavefront phase and regularize its associated noises in a natural way. The phase is statistically isotropic, but only approximately homogeneous; that is, the structure function coincides with Eq. (1) only when r⬘ and r are nearly collinear. Even with this deficiency this model is good enough to derive extended expressions for the Strehl ratio and the angle-of-arrival variance [13], but not as good as to describe more-complex quantities. Therefore, in a situation where noticeable deviations from the collinearity are found, this model fails. The problem is that the model is still based in a onedimensional fBm. A real process defined on the pupil plane must be used to obtain an exact expression for the structure function of the turbulent wavefront phase. The Lévy fractional Brownian field (LfBf) is a wellknown process [6] and a natural extension of the fractional Brownian motion to higher dimensions. That is, it conserves all the properties of the family of fBm © 2008 Optical Society of America
March 15, 2008 / Vol. 33, No. 6 / OPTICS LETTERS
processes: it is Gaussian, has mean zero, and has stationary increments and covariance E关B 共r⬘兲B 共r兲兴 = H
H
2 2
共储r⬘储2H + 储r储2H − 储r⬘ − r储2H兲; 共2兲
in our case both vectors, r and r⬘, belong to the pupil plane, R2—2 is the variance of the process at 共1 , 1兲. Nevertheless, the original definition has proven to be inadequate when dealing with practical situations or when used to construct a noise calculus. Recently, Herbin [15] proposed a generalized harmonizable representation for it: BH共r兲 =
1
冑2C2,H
冕
R2
共e
ir·
− 1兲
储储H+1
ˆ 共d2兲, W
共3兲
where = 共1 , 2兲 苸 R2, C2,H = ⌫共H兲⌫共1 − H兲 / 关22H⌫2共1 ˆ is the Fou+ H兲兴, ⌫共·兲 is the gamma function, and W rier transform of the Gaussian measure in R2—see [15] for more details. Now, the wavefront phase can be properly defined, over a pupil with radius R, as
共R兲 = A,HRHBH共兲,
储储 ⱕ 1,
共4兲
where the constant A,H normalizes the phase under some prescription— is the normalized pupil coordinate. From the covariance definition given in Eq. (2), the phase structure function is D关R共⬘ − 兲兴 = A2 ,HR2H储⬘ − 储2H
共5兲
for any two points in the pupil. In opposition to the structure function calculated in our previous work [13] there is no approximation in here. Originally defined by Fried [16] for the Kolmogorov case, the value of the structure function normalizing constant is obtained from maximizing the longexposure resolution, R, for a pupil diameter far greater than the Fried’s parameter. This definition can be easily extended to any state of the turbulent atmosphere—represented by a Hurst parameter different from H = 5 / 6 [13]. Since the normalized resolution is equal to the Strehl ratio and the wavefront phase is defined Gaussian, the procedure is functionally identical to the one given in our previous work. Now, let r0,H be the generalized Fried parameter, which is a function of H; then for the condition 2 Rmax = r0,H / 42f2 to be satisfied ( is the wavelength and f is the focal length of the optical system) it is necessary that −2H . A2 ,H = 22H+1⌫H共1 + 1/H兲r0,H
共6兲
Note that for the OK case A2 ,5/6 = 6.88r0−5/3, recovering the widely used phase structure constant. Alternatively, Nicholls, Boreman, and Dainty—see [5]—have given another prescription to set the constant A,H; the piston-subtracted wavefront phase variance is normalized to 1 rad2 when the diameter of the pupil is equal to the generalized Fried coherence
573
diameter. Now, from Eq. (4) we can easily determine the piston coefficient and then obtain the pistonremoved wavefront phase 1 共R兲 = A,HRH BH共兲 − BH共⬘兲d2 . 共7兲 储储ⱕ1
冋
册
冕
Because this phase has also mean zero, as it inherits this property from the LfBf definition, the averaged mean-square phase-error results equal to its second moment. Therefore, using the generalized harmonizable representation given in Eq. (3), the phase-error results ⬁ 关2J 共兲/ − 1兴2 1 1 2 2H d − A,HR 1 + H C2,H 0 2H+1
再
= A2 ,HR2H
冎
冕
⌫共2 + 2H兲 ⌫共2 + H兲⌫共3 + H兲
共8兲
.
Finally, imposing the normalizing condition this quantity should be equal to 1, and then 22H⌫共2 + H兲⌫共3 + H兲 −2H . 共9兲 r0,H A2 ,H = ⌫共2 + 2H兲 This last constant is smaller than the one obtained through the Fried’s definition, although it asymptotically approaches to this constant as H tends to one. Moreover, this difference becomes notorious as the Hurst parameter drops below 5 / 6. Boreman and Dainty’s definition gives a lower bound to the normalized resolution even in the asymptotic limit—it never reaches one for large pupil diameters. From now on, we will indistinctly refer to both numerical factors ac−2H as C2 ,H. companying r0,H More-complex quantities are prone to be investigated under the Lévy fractional Brownian field model. For now, let us revisit the angle of arrival, which was previously estimated under the ifBm model. Without any artificial assumptions, from the corrugation definition, it is −2H ␣共r兲 = − ⵜ 共r兲 = − Cz,Hr0,H ⵜ BH共r兲, 共10兲 2 where Cz,H = 共 / 2兲C,H. The gradient of the LfBf is defined in a properly constructed dual space of the original Gaussian stochastic processes: the fractional Hida distribution space. As we have shown [13], these “noises” revert back to a stochastic process if we convolute them with a fast decaying function, r = 共·−r兲. This function is physically real; that is, it corresponds to any aperture weighting function. Then any statistically relevant quantity can be evaluated; for example, the angle-of-arrival variance results 2 ˆ 共兲兩2 22Cz,H 兩 2 2 m,H共r兲 = E储␣共r兲储 = 2H d 2 , r0,HC2,H R2 储储2H
冕
共11兲 ˆ is the Fourier transform of the aperture where weighting function. Observe, that the angle of arrival
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is, indeed, statistically homogeneous and then independent of the point of evaluation, r. Let us just evaluate this variance over an aperture of diameter D for some optical system, e.g., a telescope. Our problem has cylindrical symmetry, and ˆ 共兲 = J1共D / 2兲 / 共D / 2兲. The variance results thus
冉 冊 1
4H⌫ H + 2 m,H 共 r兲 =
2 C2 ,H
1/2⌫共2 + H兲 22
−2H 2D2H−2r0,H . 共12兲
When we observe a beam that has traveled through a 2 2 = 0.9738m , Kolmogorov turbulence we have m,5/6 2 where m is the variance given by Tatarsk [17]. The total values of the angle-of-arrival variance coefficient are below any previously reported estimate for the Kolmogorov case: 0.340 or 0.329 using the Fried’s or Boreman’s prescription, respectively, compared with 0.342 [4], 0.358 [18], and 0.365 [19]. This result was unexpected; this deviation can be attributed to either the definition of the generalized Fried parameter r0,H, because of the imposed normalizations, or the absence of outer and inner scales in our model. Besides, if the wavefront phase should be univocally defined, then the value of the generalized Fried parameter must differ whether we are using either of the two normalizing definitions; moreover, it must depend on the Hurst parameter and thus on the dynamic state of the turbulence [2]. Under our model the coherence diameter can be experimentally retrieved using any of the well-known techniques used today. In particular, the seeing can be evaluated from the wavefront phase presented here; thus, observational data can be used to determine a value for this generalized Fried parameter. On the other hand, the outer and inner scales can be defined under a multiscale extension of the LfBf process following a similar approach to the one given by Bardet and Bertrand [20] to the multiscale (one-dimensional) fractional Brownian motion. We plan to study both directions in future works. Finally, based on the Lévy fractional Brownian field, the wavefront phase not only gives the correct structure function for non-Kolmogorov turbulence but also adds well-known statistic properties—all of them can be evaluated analytically. In contrast to our
previous model, any quantity related to the wavefront can be evaluated without limitations. This work was supported by Comisión Nacional de Investigación Científica y Tecnológica (CONICYT, FONDECYT grant 11060512, Chile), partially by Pontificia Universidad Católica de Valparaíso (grant 123.788/2007, Chile) and Consejo Nacional de Investigaciones Científicas y Técnicas (Argentina). References 1. R. G. Buser, J. Opt. Soc. Am. 61, 488 (1971). 2. M. Bester, W. C. Danchi, C. G. Degiacomi, L. J. Greenhill, and C. H. Townes, Astrophys. J. 392, 357 (1992). 3. D. Dayton, B. Pierson, B. Spielbusch, and J. Gonglewski, Opt. Lett. 17, 1737 (1992). 4. D. S. Acton, Appl. Opt. 34, 4526 (1995). 5. T. W. Nicholls, G. D. Boreman, and J. C. Dainty, Opt. Lett. 20, 2460 (1995). 6. G. Samorodnitsky and M. S. Taqqu, Stable NonGaussian Random Processes: Stochastic Models with Infinite Variance, Stochastic Modeling (Chapman & Hall/CRC 1994). 7. C. Schwartz, G. Baum, and E. N. Ribak, J. Opt. Soc. Am. A 11, 444 (1994). 8. E. N. Ribak, E. Gershnik, and M. Cheselka, Opt. Lett. 21, 435 (1996). 9. A. Belmonte, A. Comeron, J. A. Rubio, J. Bara, and E. Fernandez, Appl. Opt. 36, 8632 (1997). 10. C. Dessenne, P. Y. Madec, and G. Rousset, Appl. Opt. 37, 4623 (1998). 11. D. R. McGaughey and G. J. M. Aitken, J. Opt. Soc. Am. A 14, 1967 (1997). 12. R. A. Johnston and R. G. Lane, Appl. Opt. 39, 4761 (2000). 13. D. G. Pérez, L. Zunino, and M. Garavaglia, J. Opt. Soc. Am. A 21, 1962 (2004). 14. R. J. Elliott and J. van der Hoek, Math. Finance 13, 301 (2003). 15. E. Herbin, Rocky Mountain J. Math. 36, 1249 (2006). 16. D. L. Fried, J. Opt. Soc. Am. 56, 1372 (1966). 17. V. I. Tatarsk, Wave Propagation in a Turbulent Atmosphere (Nauka, 1967). [In Russian; English translation: The Effect of the Turbulent Atmosphere on Wave Propagation (NTIS, 1971)]. 18. M. Sarazin and F. Roddier, Astron. Astrophys. 227, 294 (1990). 19. S. S. Olivier, C. E. Max, D. T. Gavel, and J. M. Brase, Astrophys. J. 407, 428 (1993). 20. J.-M. Bardet and P. Bertrand, J. Time Ser. Anal. 28, 1 (2007).
