Temporal properties of Zernike modes for non-Kolmogorov turbulence

V. V. Voitsekhovich; V. G. Orlov

doi:10.1364/OE.24.016123

1. Introduction

During last decades the effects of non-Kolmogorov turbulence on the propagations of light waves attract a growing attention of researchers [1–14]. A number of theoretical investigations have been inspired due to the experimental data [1–4, 8–10] obtained by different techniques (stellar scintillations, balloon measurements, radar and lidar measurements, etc.) which show that the atmospheric turbulence behavior may deviates from the predictions of the Kolmogorov theory. A generalization of Kolmogorov model for weak-turbulence conditions for the case of extended power law exponent is suggested in [5]. The effect of non-Kolmogorov turbulence on the Strehl ratio and the relationship between the index structure function and the index power spectrum is investigated in [6]. The influence of non-Kolmogorov turbulence on the variances of Zernike modes is performed in [7]. The temporal statistics of irradiance and angle-of-arrival fluctuations are investigated in [11–14]

In this paper we consider the influence of non-Kolmogorov turbulence on the temporal properties of Zernike modes. This study can be useful for practical applications such as adaptive optics or imaging through the turbulence. Also because Zernike expansion is a powerful tool for theoretical investigations, the results of this study can be of interest from theoretical viewpoint.

The present investigation is based on Taylor’s frozen hypothesis. The behavior of the temporal correlation functions of Zernike modes and their dependence on the spectral index is analyzed. The calculations are performed for the case of constant turbulence parameters along the propagation path. The effects of both outer and inner scale of the turbulence are included into the study. Although the corresponding theoretical development is, in general, known [15–17], however, as it will be seen through the paper, there are certain effects of non-Kolmogorov turbulence which can be interesting from both theoretical and practical viewpoint. Mainly these effects appear due to the variations of spectral index. Also there are of interest the effects related to the combined influence of varying spectral index and the outer scale of the turbulence on temporal properties of Zernike modes. So the main attention in this investigation will be paid to the study and physical interpretation of these phenomena.

2. Theoretical calculations

The temporal correlation functions of Zernike modes for the case of light wave propagation through the Kolmogorov turbulence have been calculated in [15], while more general results related to the temporal spectra of Zernike modes are presented in [17, 18]. In this Section, following the development used in [15], we will generalize the results for the case of non-Kolmogorov turbulence and present some analytical derivations which can be useful for applications. Although the theoretical development is quite similar to that one developed in [15], we will repeat here its important points for self-consistency and clarity of this paper.

Let $S (\vec{ρ}, t)$ be a spatio-temporal turbulence-distorted phase at the aperture of diameter $D$ . This spatial part of this phase can be expanded over the two-dimensional Zernike polynomials $Z_{l}$ as follows:

S (\vec{ρ}, t) = \sum_{n = 1}^{n = \infty} a_{l} (t) Z_{l} (\vec{ρ}),

where

Z_{l}

are the two-dimensional Zernike polynomials [18] which can be written in a compact form as:

Z_{l} (ρ, φ) = \sqrt{(n + 1) (2 - δ_{0 m})} R_{n}^{m} (\frac{2 ρ}{D}) \cos {m φ + \frac{π}{4} [{(- 1)}^{l} - 1] (1 - δ_{0 m})},

where

ρ

and

φ

are the polar coordinates of the vector

\vec{ρ},

R_{n}^{m}

stands for the radial Zernike polynomials, and

δ_{0 m}

denotes the Kronecker delta-symbol.

In order to study the temporal properties of Zernike terms we will use the temporal correlation functions $B_{l} (τ)$ of Zernike coefficients $a_{l}$ . Taking into account the orthogonality of Zernike polynomials we can write $B_{l} (τ)$ as:

B_{l} (τ) = 〈 a_{l} (t) a_{l} (t + τ) 〉 = \int_{G} d^{2} ρ_{1} d^{2} ρ_{2} B_{S} (\vec{ρ_{1}}, \vec{ρ_{2}}, τ) Z_{l} (\vec{ρ_{1}}) Z_{l} (\vec{ρ_{2}}),

where

τ

is the time lag,

〈 \cdot \cdot \cdot 〉

denotes the ensemble averaging,

B_{S}

stands for the spatio-temporal correlation function of phase at the aperture, and the integration is performed over the aperture.

According to Tatarski [19] and Taylor's frozen-turbulence hypothesis the function $B_{S}$ for the initially plane light wave passed the distance $L$ through the turbulent atmosphere can be expressed as:

B_{S} (\vec{ρ_{1}}, \vec{ρ_{2}}, τ) = π k^{2} \int_{0}^{L} d z \int d^{2} κ Φ_{n} (κ, z) \exp {- i \vec{κ} (\vec{ρ_{1}} - \vec{ρ_{2}} - \vec{υ} τ)},

where

k

is the wavenumber,

Φ_{n}

denotes the spectrum of refractive-index fluctuations, and

\vec{υ}

is the two-dimensional projection of the three-dimensional wind velocity vector on the aperture.

For the case of non-Kolmogorov turbulence the spectrum $Φ_{n}$ which takes into account the influence of both inner and outer scale of the turbulence can be expressed as [20]:

\begin{array}{l} Φ_{n} (κ) = A (α) \cdot {\tilde{C}}_{n}^{2} \cdot {(κ^{2} + κ_{0}^{2})}^{- α / 2} \exp (- \frac{κ^{2}}{κ_{m}^{2}}), 3< α < 4, \\ A (α) = \frac{1}{4 π^{2}} Γ (α - 1) \cos (\frac{α π}{2}), \\ α is the spectral index, {\tilde{C}}_{n}^{2} is a generalized structure parameter, \\ κ_{0} = \frac{2 π}{L_{0}}, L_{0} is the outer scale of the turbulence, \\ κ_{m} = \frac{c (α)}{l_{0}}, l_{0} is the inner scale of the turbulence, \\ c (α) = {[Γ (\frac{5 - α}{2}) \cdot A (α) \cdot \frac{2}{3} π]}^{\frac{1}{α - 5}} . \end{array}

Assuming the constant turbulence parameters $\vec{υ}$ , ${\tilde{C}}_{n}^{2}$ , $L_{0}$ , and $l_{0}$ along the propagation path, substituting (2), (4), and (5) into (3) and performing the integrations one can get the following expression for the normalized temporal correlation functions $b_{l} (τ) = B_{l} (τ) / B_{l} (0)$ of Zernike modes:

\begin{array}{l} b_{l} (τ) = \frac{1}{N} \int_{0}^{\infty} d κ κ^{- 1} {(κ^{2} + \frac{D^{2}}{4} κ_{0}^{2})}^{- α / 2} e x p (- \frac{4 κ^{2}}{D^{2} κ_{m}^{2}}) J_{n + 1}^{2} (κ) \times \\ [J_{0} (\frac{2 υ τ}{D} κ) + {(- 1)}^{l + m} (1 - δ_{0 m}) J_{2 m} (\frac{2 υ τ}{D} κ) \cos (2 m θ)], \\ N = \int_{0}^{\infty} d κ κ^{- 1} {(κ^{2} + \frac{D^{2}}{4} κ_{0}^{2})}^{- α / 2} e x p (- \frac{4 κ^{2}}{D^{2} κ_{m}^{2}}) J_{n + 1}^{2} (κ), \end{array}

where

κ_{0}

and

κ_{m}

are related to the outer and inner scale of the turbulence through Eq. (5),

D

is the aperture diameter,

J_{k}

stands for the Bessel function of k-th order,

υ

and

θ

are the wind velocity and direction, respectively, and

δ_{0 m}

denotes the Kronecker delta-symbol.

For the case of infinite outer scale and zero-valued inner scale of the turbulence the correlation functions $b_{l} (τ)$ can be expressed analytically in terms of generalized hypergeometric functions using the following formula:

\begin{array}{l} \int_{0}^{\infty} d κ κ^{- α} J_{n + 1}^{2} (a κ) J_{l} (b κ) = \\ 2^{- α} a^{α - 1} {(\frac{b}{a})}^{l} \frac{Γ (α - l) Γ (n + \frac{l - α + 3}{2})}{Γ (n + \frac{α - l + 3}{2}) Γ (\frac{α - l + 1}{2}) Γ (l + 1)} {}_{3}F_{2} (n + \frac{l - α + 3}{2}, n + \frac{l - α - 1}{2} - n, \frac{l - α + 1}{2}; \frac{l - α + 2}{2}, l + 1; \frac{b^{2}}{4 a^{2}}) + \\ 2^{- l} a^{- 1} b^{α} \frac{Γ (l - α)}{Γ (\frac{l + α + 2}{2}) Γ (\frac{l - α + 1}{2}) Γ (\frac{1}{2})} {}_{3}F_{2} (n + \frac{3}{2}, - n - \frac{1}{2}, \frac{1}{2}; \frac{α - l + 2}{2}, \frac{α + l + 2}{2}; \frac{b^{2}}{4 a^{2}}), b < 2 a \\ \int_{0}^{\infty} d κ κ^{- α} J_{n + 1}^{2} (a κ) J_{l} (b κ) = \\ 2^{- α} b^{α - 1} {(\frac{a}{b})}^{2 n + 2} \frac{Γ (n + \frac{l - α + 3}{2})}{Γ^{2} (n + 2) Γ (\frac{l + α - 1}{2} - n)} {}_{3}F_{2} (n + \frac{3}{2}, n + \frac{l - α + 3}{2}, n + \frac{3 - l - α}{2}; 2 n + 3, n + 2; \frac{4 a^{2}}{b^{2}}), b > 2 a \end{array}

where

Γ

denotes the gamma-function, and

{}_{3}F_{2}

stands for the generalized hypergeometric function.

For the case of finite outer scale and zero-valued inner scale (or for the case of non-zero-valued inner scale and infinite outer scale) the correlation functions $b_{l} (τ)$ can be expressed analytically in terms of generalized hypergeometric functions of two arguments (the corresponding formulae allowing to evaluate the integrals of interest can be found in [21]).

3. Results

As one can see from Eq. (6) the normalized correlation function $b_{l} (τ)$ of l-th Zernike term depends on the following dimensionless parameters: $\frac{υ τ}{D}, θ, \frac{L_{0}}{D}, \frac{l_{0}}{D}, and α$ . Analyzing the dependence on the wind direction $θ$ one can see that it is determined by the second term in Eq. (6) and affects the non-symmetrical aberrations with $m \neq 0$ only, for example, the tilt $(l = 2, 3, n = 1, m = 1)$ or the astigmatism $(l = 5, 6, n = 2, m = 2)$ . Each of these non-symmetric aberrations consists of two terms (for example, tilt X and tilt Y) and, if we sum up the corresponding pair of correlations functions, the dependence on the wind direction $θ$ will disappear. From the physical viewpoint such an approach allows one to consider averaged temporal properties of a complete aberration. Also, as one can see from Eq. (6), in this case all the correlation functions for a given order of aberration $n$ will be equal. In what follows we use this wind-direction-free approach to simplify the presentation of the results.

Figure 1 shows the temporal correlation functions of the first four orders of aberrations for different magnitudes of the spectral index $α$ . Analyzing the graphs in Fig. 1 we can arrive to the following conclusions.

Fig. 1 Temporal correlation functions of Zernike modes for the first four orders of aberrations. The outer scale of the turbulence is big compared to the aperture diameter (L₀/D = 1000). The inner scale of the turbulence is small compared to the aperture diameter (l₀/D = 10⁻³). Three cases of spectral index α are presented: solid line – α = 3.1, dashed line - α = 11/3 (Kolmogorov model), dotted line - α = 3.9.

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1. The correlation time is decreasing with increasing of the order of aberration. This happens because the completeness of an aberration is growing up with increasing of its order. Simplifying, we can say that the aberration shape contains more and more extrema with increasing of the aberration order. However the more extrema are passing through the aperture during a certain time, the shorter is the correlation time of an aberration.
2. The correlation time is growing up with increasing of spectral index $α$ . The refractive index spectrum is going down faster with increasing of spectral index, i.e. the influence of low spatial frequencies is getting stronger while the effect of high ones is diminished. From the physical viewpoint it means that the number of big-sized turbulence eddies is growing up while the number of small-sized ones is decreasing. However the smaller is the number of small-sized eddies, the less chaotic is a process of light propagation that explains why the correlation time is growing up with increasing of spectral index.
3. The higher is the order of aberration, the smaller is the difference between the Kolmogorov and non-Kolmogorov turbulence. From the physical viewpoint this effect can be interpreted as follows. The geometric complicity of an aberration is growing up with its order, and the more complicate is an aberration, the stronger it is affected by small-sized turbulence eddies. As a consequence the stronger is the influence of small-sized eddies, the more chaotic is a process. However the more chaotic is a process, the less sensitive are its statistical properties to the process parameters. In our case this parameter is a variance of spectral index.

Figure 1 presents the results for the case when the outer scale of the turbulence is big compared to the aperture size (L₀/D = 1000). To analyze the influence of the outer scale we present in Fig. 2 an opposite case when the outer scale is in order of the aperture size (L₀/D = 1). Comparing the graphs in Fig. 2 we can see that in the case of small outer scale the results depend very slightly on the magnitude of the spectral index $α$ , while for the case of big outer scale (Fig. 1) this dependence is more noticeable (especially for the first order of aberrations, when $n = 1$ ). In another words, we can say that the difference between the Kolmogorov and non-Kolmogorov turbulence is getting smaller with decreasing of the outer scale magnitude. It happens because when the outer scale magnitude is decreasing, the influence of small-sized inhomogeneities is growing up making the random process more and more chaotic. However the more chaotic is a process, the weaker its statistical properties are affected by variance of parameters, in our particular case, by variance of the spectral index.

Fig. 2 Temporal correlation functions of Zernike modes for the first four orders of aberrations. The outer scale of the turbulence equal to the aperture diameter (L₀/D = 1). The inner scale of the turbulence is small compared to the aperture diameter (l₀/D = 10⁻³). Three cases of spectral index α are presented: solid line - α = 3.1, dashed line - α = 11/3 (Kolmogorov model), dotted line - α = 3.9.

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In more details the influence of the outer scale on the temporal properties of Zernike modes is illustrated in Fig. 3 where we show how the correlation time depends on the outer scale. The correlation time is defined as the width of a correlation function on its half height and shows how fast the corresponding aberration is changing with time. Figure 3 presents the correlation time vs the outer scale for the first four orders of aberrations. One can see that the effect of the outer scale is quite strong for low-order aberrations ( $n = 1$ ), while for the high-order ones it starts to be noticeable only when the outer scale magnitude is approaching to the aperture size.

Fig. 3 Correlation time of Zernike modes vs outer scale of the turbulence for the first four orders of aberrations. The inner scale of the turbulence is small compared to the aperture diameter (l₀/D = 10⁻³). Three cases of spectral index α are presented: solid line - α = 3.1, dashed line - α = 11/3 (Kolmogorov model), dotted line - α = 3.9.

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Figure 4 illustrates the effect of inner scale of the turbulence on the temporal properties of Zernike modes. In this figure we plot the correlation time vs the inner scale magnitude for the first four orders of aberrations. One can see that the correlation time is growing up with the inner scale increment. It happens because the contribution of high spatial frequencies is getting down with the increasing of inner scale that makes the process less chaotic. For the same reason the correlation time is increasing when the spectral index is growing up. However these inner-scale-related effects start to be more or less pronounced when the inner scale magnitude is approaching to the aperture size i.e. when the aperture size is very small.

Fig. 4 Correlation time of Zernike modes vs inner scale of the turbulence for the first four orders of aberrations. The outer scale of the turbulence is big compared to the aperture diameter (L₀/D = 1000). Three cases of spectral index α are presented: solid line - α = 3.1, dashed line - α = 11/3 (Kolmogorov model), dotted line - α = 3.9.

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4. Conclusions

Temporal properties of Zernike modes for the case of non-Kolmogorov turbulence have been investigated. The calculations have been performed involving the Taylor's frozen-turbulence hypothesis for the case of weak-turbulence conditions with constant turbulence parameters along the propagation path. The effects of both the inner and the outer scale of the turbulence have been included into the study.

To study the properties of interest the behavior of the temporal correlation functions of Zernike modes has been analyzed. The analysis of the results obtained allows one to arrive to the following conclusions:

- the higher is the order of aberration, the smaller is the difference between the Kolmogorov and non-Kolmogorov turbulence;
- the smaller is the outer scale magnitude, the smaller is the difference between the Kolmogorov and non-Kolmogorov turbulence;
- the correlation time is growing up with increasing of spectral index $α$ ;
- the correlation time is decreasing with increasing of the order of aberration;
- the effects of the inner scale are weak and they are getting more or less pronounced when the inner scale magnitude is approaching to the aperture size.

Acknowledgments

This work was supported by Dirección General de Asuntos del Personal Académico (UNAM, México) under the projects IN102514 and IT101116 (PAPIIT).

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Temporal properties of Zernike modes for non-Kolmogorov turbulence

Abstract

1. Introduction

2. Theoretical calculations

3. Results

4. Conclusions

Acknowledgments

References and Links

Cited By

Figures (4)

Equations (7)

Optics Express