Equivalent refractive-index structure constant of non-Kolmogorov turbulence

Yujie Li; Wenyue Zhu; Xiaoqing Wu; Ruizhong Rao

doi:10.1364/OE.23.023004

1. Introduction

For a long time, Kolmogorov theory has been used to study the optical wave propagation through atmospheric turbulence [1–4]. However, with the development of measurement techniques and the increasing number of experimental researches for atmospheric turbulence, more and more measurement results indicate that atmospheric turbulence has not always obey the characteristics of Kolmogorov statistics [5–8]. The non-Kolmogorov turbulence is proposed, whose ideal model of three-dimensional power spectrum has the features: the spectral power law $α$ takes a value ranging from 3 to 4 rather than the Kolmogorov spectrum’s 11/3,the amplitude factor is a function of spectral power law rather than 0.033, most important parameter is the structure constant with units $m^{3 - α}$ rather than $m^{- 2 / 3}$ , where m is a unit of length, refers to meter [9]. Using the non-Kolmogorov spectrum, many theoretical researches on optical wave propagation in non-Kolmogorov atmospheric turbulence have been developed, and some theoretical results have been reported [9–14]. However, so far almost all of the experimental data of the structure constant are mainly based on the Kolmogorov theory. In addition, the non-Kolmogorov structure constant has a varying dimension, which leads to many difficulties in describing the statistical properties of wave propagation in turbulent atmosphere [15–18]. Therefore, it is very necessary to find out the relationship between the Kolmogorov structure constant and the non-Kolmogorov structure constant in order to make a comprehensive understanding of optical wave propagating through atmospheric turbulence.

At present, there are two equivalent non-Kolmogorov structure constant models. One is presented by Yahya Baykal and Hamza Gerçekcioğlu [19]. The way to acquire the equivalent structure constant is that they let the spherical wave scintillation index in non-Kolmogorov turbulence and that in Kolmogorov turbulence equal in horizontal path. The other equivalent structure constant is proposed by Arkadi Zilberman et al [12]. The way to obtain the structure constant is that they let the non-Kolmogorov spectrum and the Kolmogorov spectrum equal at the Fresnel scale $\sqrt{L λ}$ for a wavelength $λ$ and a path length L. Both of the equivalent structure constant models depend on the parameters of optical wave propagation as well as the turbulent characteristic parameters. Because the first equivalent relation is based on the spherical wave scintillation in weak turbulent atmosphere, it may not be used to investigate all kinds of wave and turbulent effects and the propagation in strong fluctuations region. The second equivalent relation is based on the turbulent spectrum in Fresnel scale which mainly has effects on intensity fluctuations. We know that in weak turbulence, the correlation length of intensity fluctuations has the size of a Fresnel scale, however, in strong turbulence the correlation length is determined by coherent length. So it may also not be used to investigate phase fluctuations effects and intensity fluctuations effects in strong turbulent region. It is necessary to notice that structure constant is used to describe the strength of turbulence fluctuations. It should have relations to the parameters of turbulence itself only, rather than optical propagation parameters such as wavelength and optical path length. Both of the above two equivalent structure constants are non-objective parameters, which contain not only the parameters of turbulence itself such as the power law, the Kolmogorov structure constant but also the path length and the wavelength. Thus they are not suitable for describing the turbulent strength.

In this paper, based on the refractive-index structure functions and the variances of refractive-index fluctuations of atmospheric turbulence, the expression of the equivalent structure constant of non-Kolmogorov turbulence is derived. And then the relations of the non-Kolmogorov structure constant, the Kolmogorov structure constant, the outer scale and the power law are analyzed. Finally, comparisons of the equivalent results and the experimental results of non-Kolmogorov structure constants are illustrated.

2. Kolmogorov and non-Kolmogorov structure function

By using dimensional analysis, Kolmogorov showed that the longitudinal structure function of wind velocity in the inertial range satisfied the universal 2/3 power law [1]

D_{v} (\vec{r}) = 〈 {[v ({\vec{r}}_{1} + \vec{r}) - v ({\vec{r}}_{1})]}^{2} 〉 = C_{v}^{2} {\vec{r}}^{2 / 3}, l_{0} ≪ r ≪ L_{0},

where

〈 \cdot 〉

denotes ensemble average,

v ({\vec{r}}_{1})

is the velocity vector at point

{\vec{r}}_{1}

,

\vec{r}

is a position vector, the scalar

r

is the magnitude of the vector

\vec{r}

, l₀ and L₀ are the inner scale and outer scale, respectively,

C_{v}^{2}

is the wind velocity vector structure constant.

The statistical description of the turbulence-induced random fluctuations in the atmospheric refractive-index is similar to that for the related random field of turbulent velocities. For statistically homogeneous and isotropic Kolmogorov turbulence, Tatarskii derived the refractive-index structure function in inertial range [2]

D_{n} (r) = 〈 {[n (r_{1} + r) - n (r_{1})]}^{2} 〉 = C_{n}^{2} r^{2 / 3}, l_{0} ≪ r ≪ L_{0},

where

D_{n} (r)

is the structure function of the refractive-index,

n (r_{1})

is the refractive-index at the point r₁,

C_{n}^{2}

is called refractive-index structure constant and has units of

m^{- 2 / 3}

.

By using the Weiner-Khintchin theory, Tatarskii derived the relations of the refractive-index structure function and the power spectrum density function in turbulence [2]

D_{n} (r) = 8 π \int_{0}^{\infty} d κ Φ_{n} (κ) κ^{2} [1 - \frac{\sin (κ r)}{κ r}],

Φ_{n} (κ) = \frac{1}{4 π^{2} κ^{2}} \int_{0}^{\infty} \frac{\sin (κ r)}{κ r} \frac{d}{d r} [r^{2} \frac{d}{d r} D_{n} (r)],

where

Φ_{n} (κ)

is the power spectrum of turbulence,

κ

is the spatial wave number.

Substituting the refractive-index structure function Eq. (2) into Eq. (4) and evaluating the derivatives yields the three-dimensional Kolmogorov turbulent power spectrum [2]

Φ_{n} (κ, z) = 0.033 C_{n}^{2} (z) κ^{- 11 / 3}, 2 π / L_{0} ≪ κ ≪ 2 π / l_{0},

where

Φ_{n} (κ, z)

is the turbulent spectrum along the optical path z,

C_{n}^{2} (z)

is the refractive-index structure constant along the optical path z.

For non-Kolmogorov turbulence, the structure function doesn’t satisfy 2/3 power law in inertial range, it is described by [9]

D_{n} (r, z) = {\tilde{C}}_{n}^{2} (γ, z) r^{γ}, l_{0} ≪ r ≪ L_{0},

where

{\tilde{C}}_{n}^{2} (γ, z)

is the non-Kolmogorov structure constant along the optical path z and has units of

m^{- γ}

which is similar to

C_{n}^{2} (z)

,

γ

is the index of structure function.

And then substituting the non-Kolmogorov structure function Eq. (6) into Eq. (4) and evaluating the derivatives yields the three-dimensional non-Kolmogorov turbulent power spectrum in inertial range [9]

Φ_{n} (α, κ) = A (α) {\tilde{C}}_{n}^{2} (α, z) κ^{- α}, 2 π / L_{0} ≪ κ ≪ 2 π / l_{0},

where

α

is the spectral power law,

α \equiv γ + 3

,

A (α)

is the coefficient which maintains consistency between the structure function and its power spectrum, which is defined by [9]

A (α) = 2^{α - 6} (α^{2} - 5 α + 6) π^{- 3 / 2} \frac{Γ [\frac{3 - α}{2}]}{Γ [\frac{5 - α}{2}]}, 3 < α < 5,

where the symbol

Γ [\cdot]

denotes the gamma function. When the power law

α

approaches the limiting value 3, the function

A (α)

approaches zero. When

α = 11 / 3

,

A (11 / 3) = 0.033

, the Eq. (7) reduces to Eq. (5) and the non-Kolmogorov power spectrum reduces to the Kolmogorov power spectrum.

3. The equivalent non-Kolmogorov structure constant

The random field of turbulence fluctuations is basically nonisotropic for large separation distance $r > L_{0}$ . However, if the random field is homogeneous and isotropic even for large separation distance, according to the definition of the outer scale, when the separation distance of two points is larger than or equal to the outer scale, the refractive-index fluctuations of the two points are independent. The correlation of two points in the random field can be ignored. The structure function will asymptotically approach the value $2 σ_{n}^{2}$ , where $σ_{n}^{2}$ is the variance of the refraction-index fluctuations, the relation can be expressed as [3, 20]

\begin{array}{l} D_{n} (L_{0}, z) = 〈 {[n (r + L_{0}) - n (r)]}^{2} 〉 \\ = 〈 n^{2} (r + L_{0}) 〉 + 〈 n^{2} (r) 〉 - 2 〈 n (r + L_{0}) 〉 〈 n (r) 〉 \\ = 2 σ_{n}^{2} = C_{n}^{2} (z) L_{0}^{2 / 3}, \end{array}

therefore the Kolmogorov structure constant can be expressed as a function of the variance of the refractive-index fluctuations and the outer scale

C_{n}^{2} (z) = 2 σ_{n}^{2} L_{0}^{- 2 / 3},

or the variance of the refractive-index fluctuations can be expressed as

σ_{n}^{2} = 0.5 C_{n}^{2} (z) L_{0}^{2 / 3} .

Accordingly, for Kolmogorov turbulence, the structure function Eq. (2) can be expressed as

D_{n} (r, z) = 2 σ_{n}^{2} {(r / L_{0})}^{2 / 3}, l_{0} ≪ r ≪ L_{0},

similarly, the non-Kolmogorov structure function can be expressed as

D_{n} (r, z) = 2 σ_{n}^{2} {(r / L_{0})}^{3 - α}, l_{0} ≪ r ≪ L_{0},

and then by inserting the Eq. (13) into Eq. (6), the structure constant in non-Kolmogorov turbulence can be expressed as

{\tilde{C}}_{n}^{2} (α, z) = 2 σ_{n}^{2} L_{0}^{3 - α} .

Equation (14) indicates that the non-Kolmogorov structure constant can be expressed as a function of the variance of the refractive-index fluctuations, the outer scale and the spectral power law of non-Kolmogorov turbulence. It is independent of the parameters of optical wave propagation and just dependent on the characteristics of turbulence itself. By measuring the variance of refractive index fluctuations, the outer scale and the spectral power law, the structure constants can be achieved.

In order to find out the relationship between non-Kolmogorov structure constant and Kolmogorov structure constant, by inserting the Eq. (11) into Eq. (14) the non-Kolmogorov structure constant is derived

{\tilde{C}}_{n}^{2} (α, z) = L_{0}^{11 / 3 - α} C_{n}^{2} (z) .

Equation (15) is the main result of this paper, which presents the equivalent relationship between the non-Kolmogorov structure constant and the Kolmogorov structure constant.

According to the Eq. (15), we calculate the values of non-Kolmogorov structure constants changing with Kolmogorov structure constants ranging from weak to strong turbulence with different outer sales at three different power laws 3.1, 11/3, 3.9. As a result of outer scale usually has the order of meter, we set the three outer scales to 1m, 5m and 10m, and other values may be selected if necessary. The results are shown in Figs. 1(a)-1(c), respectively.

Fig. 1 The non-Kolmogorov structure constant as a function of Kolmogorov structure constant with different outer scales at three spectral power laws: (a) $α = 3.1$ , (b) $α = 11 / 3$ , (c) $α = 3.9$ .

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Figures 1(a)-1(c) show that the non-Kolmogorov structure constants are proportional to the Kolmogorov structure constants for the three power laws, and any other power laws ranging from 3 to 4. It is deduced from Figs. 1(a) and 1(c) that for a fixed Kolmogorov structure constant, with the increment of the outer scale, the non-Kolmogorov structure constant increases for the spectral power laws less than 11/3, Whereas for the spectral power law greater than 11/3, the non-Kolmogorov structure constant decreases with the increasing outer scale. Figure 1(b) shows that when the spectral power law $α = 11 / 3$ , the non-Kolmogorov structure constant is equal to the Kolmogorov structure constant for an arbitrary finite outer scale, and the non-Kolmogorov turbulence converts to the Kolmogorov turbulence.

4. Measurement methods and results comparisons

In order to verify the equivalent relation, using the Three-Dimensional Sonic Anemometers and the method of frequency spectrum analysis [8], the Kolmogorov structure constants and the non-Kolmogorov structure constants and the spectrum power laws are measured. The instruments installed on an iron tower are located at the elevations of 10m, 15m and 25m, respectively, and the underlying surface is the grass.

4.1 The methods of measuring the Kolmogorov and non-Kolmogorov structure constants and the power laws

For Kolmogorov turbulence, in the inertial range the refractive-index structure constant $C_{n}^{2}$ and the temperature structure constant $C_{T}^{2}$ are defined, respectively [3]

C_{n}^{2} = 〈 [n (\vec{x}) - n (\vec{x} + \vec{r})] 〉 r^{- 2 / 3}, l_{0} ≪ r ≪ L_{0},

C_{T}^{2} = 〈 [T (\vec{x}) - T (\vec{x} + \vec{r})] 〉 r^{- 2 / 3}, l_{0} ≪ r ≪ L_{0},

where

\vec{x}

and

\vec{r}

are the spatial position vectors, r is the value of

\vec{r}

, n is the refractive index, T is atmospheric temperature, the symbol

〈 \cdot 〉

denotes an ensemble average.

Refraction-index fluctuations associated with the visible and near-infrared region of the spectrum are due primarily to random temperature fluctuations. The relationship between the temperature structure constant $C_{T}^{2}$ and the refractive-index structure constant $C_{n}^{2}$ can be expressed as [3]

C_{n}^{2} = {(79 \times 10^{- 6} \frac{P}{T^{2}})}^{2} C_{T}^{2},

where P is atmospheric pressure with units hPa, T is atmospheric temperature with units K.

Based on the Doppler Effect and the relations of temperature and humidity as a function of sonic velocity, the Three-Dimensional Sonic Anemometer measures the transmission times of ultrasound wave pulses over a certain distance on three non-orthogonal axes. By coordinate transition, the three components of the velocity and the ultrasound temperature are obtained. The relationship between the ultrasound temperature T_s and the atmospheric temperature T can be expressed as [8]

T = \frac{T_{s}}{1 + 0.51 q},

T_{s} = \frac{c^{2}}{γ_{d} R_{d}} - 273.15,

where q denotes specific humidity, c is sonic velocity,

γ_{d}

is ratio of level pressure to level volume of dry air,

R_{d}

is gas constant of dry air.

Based on the Taylor hypothesis, it can be shown that the relationship between the one-dimensional wave number spectrum $ψ_{T} (κ)$ and the one-dimensional frequency spectrum $ψ_{T} (f)$ of temperature pulse is given by [8]

ψ_{T} (κ) = \frac{V}{2 π} ψ_{T} (f),

where f is the frequency with units s⁻¹,

f = \frac{V}{2 π} κ

,

κ

is the spatial wave number with units

m^{- 1}

, V is the wind velocity units m/s, and

ψ_{T} (f)

has following form [8]

ψ_{T} (f) = B f^{- α_{1}},

where B is the coefficient relating to the generally temperature structure constant

{\tilde{C}}_{T}^{2}

,

α_{1}

is the one-dimensional spectral power law which has the relation with three-dimensional spectral power law

α

α = α_{1} + 2.

By taking the logarithm both sides of Eq. (22), we can obtain the relation [8]

\log [ψ_{T} (f)] = \log (B) - α_{1} \log (f),

the one-dimensional spectral power law

α_{1}

of temperature fluctuations and the coefficient B can be achieved by using the linear least-squares fitting method.

For Kolmogorov turbulence, the one-dimensional power law $α_{1} = 5 / 3$ , the one-dimensional spectrum of temperature fluctuations in the inertial domain has the form [5]

ψ_{T} (κ) = 0.125 C_{T}^{2} κ^{- 5 / 3} .

By substituting the Eq. (21) into Eq. (25), the temperature structure constant $C_{T}^{2}$ can be achieved. Then by using the Eq. (18), the Kolmogorov refractive-index structure constants $C_{n}^{2} (z)$ can be obtained.

For non-Kolmogorov turbulence, by substituting $α_{1}$ into Eq. (23), the three-dimensional power law $α$ can be achieved.

The relation between the three-dimensional spectrum and the one-dimensional spectrum is given by [3]

Φ_{T} (κ) = - \frac{1}{2 π} \frac{d ψ_{T} (κ)}{d κ} .

Based on the Eq. (21) and Eq. (26), the three-dimensional temperature spectrum $Φ_{T} (κ)$ can be calculated. Because the temperature spectrum should have the same form as refractive-index spectrum Eq. (7), it can be expressed as

Φ_{T} (α, κ) = A (α) {\tilde{C}}_{T}^{2} (α, z) κ^{- α}, 2 π / l_{0} ≪ κ ≪ 2 π / L_{0},

then by substituting

α

and three-dimensional spectrum of temperature fluctuations into Eq. (27), the generally temperature structure constant

{\tilde{C}}_{T}^{2} (α, z)

can be obtained. Using the Eq. (18), the generally refractive-index structure constants

{\tilde{C}}_{n}^{2} (α, z)

for different power laws can be obtained.

4.2 The comparison of non-Kolmogorov structure constants between experimental results and equivalent results

Figure 2 is the measurements of the time series of the three-dimensional spectral power laws at the elevation of 10m. It can be seen from Fig. 2 that the spectral power laws fluctuate around the value of 11/3, similar behaviors are also found at other elevations, which proves the existence of non-Kolmogorov turbulence. Figure 3 is the comparison of measurement results of Kolmogorov structure constants and non-Kolmogorov structure constants time series at this elevation. We also can find that the Kolmogorov and the non-Kolmogorov structure constants almost have the same tendency as the non-Kolmogorov spectral power law versus the date of time, which corresponding to the linear relationship in Figs. 1(a)-1(c). Most of all, when the turbulence is very weak or the spectral power law is much less than 11/3, the value of Kolmogorov structure constant is less than non-Kolmogorov structure constant.

Fig. 2 The time series of the power laws of the three-dimensional spectrum of temperature fluctuations

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Fig. 3 The time series of measurements of Kolmogorov structure constants $C_{n}^{2} (z)$ and non-Kolmogorov structure constants ${\tilde{C}}_{n}^{2} (α, z)$ .

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Using the spectral power laws in Fig. 2 and the Kolmogorov structure constants in Fig. 3, according to the Eq. (15) we calculate the equivalent non-Kolmogorov structure constant. Figure 4 compares the measurement results of the non-Kolmogorov structure constants with the equivalent non-Kolmogorov structure constants. According to the outer scale model [21], the outer scale is about 4m at this altitude. It shows that the measurement results and the equivalent results of non-Kolmogorov structure constants coincide with each other well for almost all the power laws, as expected. It demonstrates that the method of the equivalence of non-Kolmogorov and Kolmogorov structure constants is valid.

Fig. 4 The comparisons of the time series of the non-Kolmogorov structure constants between the measurement results and the equivalent results.

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5. Conclusion

In this paper, the equivalent expression between the Kolmogorov structure constant and the non-Kolmogorov structure constant has been derived base on the structure function and the variance of refractive-index fluctuations. The results show that the non-Kolmogorov structure constant can be expressed as a function of turbulent outer scale, power law of non-Kolmogorov turbulent spectrum and Kolmogorov structure constant. And the non-Kolmogorov structure constants are proportional to the Kolmogorov structure constants, the scaling factors depend on the outer scales and the power laws. For a fixed Kolmogorov structure constant, the increment of outer scale leads to an increase in the non-Kolmogorov structure constant when power law is lower than 11/3, however, the increment of outer scale decreases the non-Kolmogorov structure constant when power law is higher than 11/3. Using this equivalent expression and the experimental data, we compared the equivalent non-Kolmogorov structure constants with the measurement results which also suggested the valid of the equivalent relation.

The results in this study provide a way of calculating the non-Kolmogorov structure constants by the measured Kolmogorov structure constants. In addition, we could still use the Kolmogorov structure constants to describe the strength of the non-Kolmogorov turbulence fluctuations. It can avoid many problems with respect to the unfixed dimensions of non-Kolmogorov structure constants when investigating the optical wave propagation in non-Kolmogorov atmospheric turbulence.

References and links

1. A. N. Kolmogorov, “The local structure of turbulence in incompressible viscous fluids for very large Reynolds numbers,” C. R. Acad. Sci. 30, 301–305 (1941).

2. V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).

3. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Medium, 2nd ed. (SPIE, 2005).

4. R. Rao, Light Propagation in the Turbulent Atmosphere (AnHui Science and Technology, 2005).

5. D. T. Kyrazis, J. B. Wissler, D. D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2120, 43–55 (1994). [CrossRef]

6. M. S. Belen’kii, S. J. Karis, J. M. Brown II, and R. Q. Fugate, “Experimental study of the effect of non-Kolmogorov stratospheric turbulence on star image motion,” Proc. SPIE 3126, 113–123 (1997). [CrossRef]

7. C. Rao, W. Jiang, and N. Ling, “Measuring the power-law exponent of an atmospheric turbulence phase power spectrum with a Shack Hartmann wave-front sensor,” Opt. Lett. 24(15), 1008–1010 (1999). [CrossRef] [PubMed]

8. X. Wu, Y. Huang, H. Mei, S. Shao, H. Huang, X. Qian, and C. Cui, “Measurement of non-Kolmogorov turbulence characteristic parameter in atmospheric surface layer,” Acta Opt. Sin. 34(6), 0601001 (2014). [CrossRef]

9. B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995). [CrossRef]

10. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007). [CrossRef]

11. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Scintillation index of optical plane wave propagating through non-Kolmogorov moderate-strong turbulence,” Proc. SPIE 6747, 67470B (2007). [CrossRef]

12. A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Propagation of electromagnetic waves in Kolmogorov and non-Kolmogorov atmospheric turbulence: three-layer altitude model,” Appl. Opt. 47(34), 6385–6391 (2008). [CrossRef] [PubMed]

13. L. Tan, W. Du, J. Ma, S. Yu, and Q. Han, “Log-amplitude variance for a Gaussian-beam wave propagating through non-Kolmogorov turbulence,” Opt. Express 18(2), 451–462 (2010). [CrossRef] [PubMed]

14. L. Cui, B. Xue, L. Cao, S. Zheng, W. Xue, X. Bai, X. Cao, and F. Zhou, “Irradiance scintillation for Gaussian-beam wave propagating through weak non-Kolmogorov turbulence,” Opt. Express 19(18), 16872–16884 (2011). [CrossRef] [PubMed]

15. H. Gerçekcioğlu and Y. Baykal, “Intensity fluctuations of flat-topped beam in non-Kolmogorov weak turbulence,” J. Opt. Soc. Am. A 29(2), 169–173 (2012). [CrossRef] [PubMed]

16. M. Charnotskii, “Common omissions and misconceptions of wave propagation in turbulence: discussion,” J. Opt. Soc. Am. A 29(5), 711–721 (2012). [CrossRef] [PubMed]

17. M. Charnotskii, “Intensity fluctuations of flat-topped beam in non-Kolmogorov weak turbulence: comment,” J. Opt. Soc. Am. A 29(9), 1838–1840 (2012). [CrossRef] [PubMed]

18. H. Gerçekcioğlu and Y. Baykal, “Intensity fluctuations of flat-topped beam in non-Kolmogorov weak turbulence: reply,” J. Opt. Soc. Am. A 29(9), 1841–1842 (2012). [CrossRef]

19. Y. Baykal and H. Gerçekcioğlu, “Equivalence of structure constants in non-Kolmogorov and Kolmogorov spectra,” Opt. Lett. 36(23), 4554–4556 (2011). [CrossRef] [PubMed]

20. R. Rao, “Optical properties of atmospheric turbulence and their effects on light propagation (invited paper),” Proc. SPIE 5832, 1–11 (2005).

21. V. P. Lukin, “Estimation of behavior of outer scale of turbulence from optical measurements,” Proc. SPIE 4538, 74–84 (2002). [CrossRef]

Equivalent refractive-index structure constant of non-Kolmogorov turbulence

Abstract

1. Introduction

2. Kolmogorov and non-Kolmogorov structure function

3. The equivalent non-Kolmogorov structure constant

4. Measurement methods and results comparisons

4.1 The methods of measuring the Kolmogorov and non-Kolmogorov structure constants and the power laws

4.2 The comparison of non-Kolmogorov structure constants between experimental results and equivalent results

5. Conclusion

References and links

Cited By

Figures (4)

Equations (27)

Optics Express