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Angle of arrival fluctuations considering turbulence outer scale for optical waves’ propagation through moderate-to-strong non-Kolmogorov turbulence Linyan Cui,* Bindang Xue, Xiaoguang Cao, and Fugen Zhou School of Astronautics, Beihang University, Beijing 100191, China *Corresponding author: [email protected] Received January 3, 2014; revised February 17, 2014; accepted February 22, 2014; posted February 24, 2014 (Doc. ID 203734); published March 27, 2014 Based on the generalized von Kármán spectrum and the extended Rytov theory, new analytic expressions for the variance of angle of arrival (AOA) fluctuations are derived for optical plane and spherical waves propagating through moderate-to-strong non-Kolmogorov turbulence with horizontal path. They consider finite turbulence outer scale and general spectral power law value, and cover a wide range of non-Kolmogorov turbulence strength. When the turbulence outer scale is set to infinite, the new expressions can reduce correctly to previously published analytic expressions [J. Opt. Soc. Am. A, 30 2188 (2013]. The final results show that the increased turbulence outer scale value enlarges the variance of AOA fluctuations greatly under moderate-to-strong (or strong) non-Kolmogorov turbulence. © 2014 Optical Society of America OCIS codes: (010.1290) Atmospheric optics; (010.1300) Atmospheric propagation; (010.1330) Atmospheric turbulence. http://dx.doi.org/10.1364/JOSAA.31.000829

1. INTRODUCTION Atmospheric turbulence has significant degrading effects for an optical wave’s propagation due to the random atmospheric refractive index fluctuations. A series of turbulence effects are produced, including irradiance scintillation, angle of arrival (AOA) fluctuations, and so on, in which the AOA fluctuations are related to the image distortion in the focal plane of an imaging or laser system. In recent years, many researchers have focused on the nonKolmogorov turbulence case, which covers a wider range of atmospheric layers [1–6]. The methods to investigate the variance of AOA fluctuations in non-Kolmogorov turbulence are generally divided into two classes: the Rytov theory-based method and the extended Rytov theory-based method, where the former is suitable for the weak non-Kolmogorov turbulence (σ 2R ≪ 1, σ 2R is the Rytov variance). Several turbulence refractive index fluctuation spectral models, including the non-Kolmogorov model [7], the generalized von Kármán model [8], and the generalized exponential model [9], have been adopted to derive the analytic expression of AOA fluctuations under weak non-Kolmogorov turbulence [8,10,11]. The extended Rytov theory-based method is developed for the moderate-to-strong non-Kolmogorov turbulence (σ 2R ≫ 1). Based on this theory, the effective non-Kolmogorov spectral model [12] has been applied to derive the analytic expressions of variance of AOA fluctuations [13], and they reduced to the results derived for the weak non-Kolmogorov turbulence in the limiting case of σ 2R ≪ 1. However, they have not considered the influences of finite turbulence inner and outer scales. As the AOA fluctuations are caused mainly by the large-scale turbulence cells, the turbulence outer scale plays an important role in the analysis [14]. 1084-7529/14/040829-07$15.00/0

Therefore, in this study, based on the extended Rytov theory, the non-Kolmogorov atmospheric refractive index fluctuations spectral model, which is derived from the generalized exponential spectral model, a new model will be introduced to investigate the variance of AOA fluctuations under moderate-to-strong non-Kolmogorov turbulence. Numerical calculations are conducted to analyze general spectral power law, turbulence outer scale, and the turbulence strength’s influences on the final expressions.

2. EFFECTIVE NON-KOLMOGOROV SPECTRUM FOR MODERATE-TO-STRONG NON-KOLMOGOROV TURBULENCE The effective non-Kolmogorov turbulence spectrum for moderate-to-strong non-Kolmogorov turbulence takes the form as [12] Φn1 κ; α  Φn κ; αGκ; α; 2π∕L0 ≪ κ ≪ 2π∕l0 ; 3 < α < 4; (1) where Φn κ; α is the non-Kolmogorov turbulence spectral model derived for weak non-Kolmogorov turbulence, given by Φn κ; α  Aα · Cˆ 2n · κ −α ; 2π∕L0 ≪ κ ≪ 2π∕l0 ; 3 < α < 4; (2)

Aα 

  Γα − 1 π ; sin α − 3 2 4π 2

© 2014 Optical Society of America

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where α is the spectral power law, Cˆ 2n is the generalized refractive index structure constant (with units of m3−α , when α  11∕3, with units of m−2∕3 ), κ is the spatial wave number with units of rad∕m and is related to the size of turbulence cells, and L0 and l0 are the turbulence outer scale and inner scale, respectively. Γ· is the gamma function. Gκ; α takes the form Gκ; α  GX κ; α  GY κ; α;

(4)



κ2 GX κ; α  exp − 2 ; κ X α

κα : GY κ; α  2 κ  κ2Y αα∕2 (5)

Here, κX α is a large-scale (or refractive) spatial-frequency cutoff, and κY α is a small-scale (or diffractive) spatialfrequency cutoff. Gκ; α acts like a spatial filter function that permits only low-pass spatial frequencies κ < κX α or highpass spatial frequencies κ > κY α to influence the wave propagation. With the turbulence inner and outer scales taken into consideration, the large-scale and small-scale spatial filters become [15]  2 κ GX κ; l0 ; L0 ; α  f κ; l0 ; αgκ; L0  exp − 2 ; κX κα GY κ; α  2 : (6) κ  κ2Y αα∕2 Here, f κ; l0 ; α describes inner scale modifications of the basic non-Kolmogorov spectrum and gκ; L0  describes turbulence outer scale effects. f κ; l0 ; α  exp−κ2 ∕κ2m , gκL0   1 − exp−κ2 ∕κ20 , and κ 0  8π∕L0 , are chosen for mathematical analysis [15]. In this case, the large-scale filter function takes the form  2  2   2  κ κ κ GX κ; l0 ; L0 ; α  exp − 2 exp − 2 − exp − 2 : κm κX κ X0

Dωp ρ  8π 2 k2

Dωs ρ 

8π 2 k2 L

Z

3. VARIANCE AOA FLUCTUATIONS FOR MODERATE-TO-STRONG NONKOLMOGOROV TURBULENCE With the geometrical optics method, the variance of AOA fluctuations of an optical wave at the receiver plane can be described in terms of the phase structure function [14]:

Z

L

dz

0

Z

∞ 0

Z

1



0



0

1 − J 0 κρΦn1 κκdκ;

(10)

1 − J 0 κρξΦn1 κκdκ:

(11)

Here, Dspl ρ and Dssp ρ are the phase structure functions for the plane and spherical wave, respectively. Φn1 κ is the effective turbulence spectral model for a wave propagating through moderate-to-strong Kolmogorov turbulence. J 0 · is the Bessel function of zero order. To consider the influences of general spectral power law values and turbulence outer scale values, in the next section, Dspl ρ, Dssp ρ and Φn1 κ will be replaced by Dspl ρ; L0 ; α, Dssp ρ; L0 ; α and Φn1 κ; L0 ; α, respectively, and theoretical expressions of variance of AOA fluctuations will be obtained for moderate-to-strong non-Kolmogorov turbulence. A. Variance AOA Fluctuations for Plane Wave under Moderate-to-Strong Non-Kolmogorov Turbulence Based on the extended Rytov theory, the wave structure function for plane wave propagating through moderate-to-strong non-Kolmogorov turbulence takes the form as Dωp ρ; L0 ; α 

8π 2 k2

Z

Z

L

0

dz



0

1 − J 0 κρΦn1 κ; L0 ; ακdκ: (12)

Here, J 0 is the Bessel function of the first kind and zero order and L is the optical path. Expanding J 0 in the form of a Maclaurin series [18] yields

(7) Here, κ2X0  κ2X κ20 ∕κ2X  κ20 . As the AOA fluctuations are caused mainly by large-scale turbulence cells [14], here only GX κ; l0 ; L0 ; α be kept in the analysis and turbulence inner scale’s influence can be ignored. At this time, the effective non-Kolmogorov spectrum, which considers finite turbulence outer scale, can be approximated as   2  2  κ κ Φn1 κ; L0 ; α ≈ Aα · Cˆ 2n · κ−α exp − 2 − exp − 2 : κX κX0 (8)

(9)

Here, Dω · is the phase structure function with the radial distance ρ  D, D is the diameter of aperture receiver, k  2π∕λ, and λ is the optical wavelength. For moderate-to-strong Kolmogorov turbulence, Dω · takes the form as [16,17]

where GX κ; α and GY κ; α are the large-scale filter and small-scale filter, respectively: 

Dω D : kD2

hβ2a i 

J 0 x 

 2n −1n x ; · 2 n! · Γn  1 n0

∞ X

(13)

where Γn  1 in the above equation can be expressed with Γn  1  n!  1n . The wave structure function for the plane wave becomes Dωp ρ; L0 ; α  8π 2 k2 AαCˆ 2n  2n Z L X ∞ −1n−1 ρ × · 2 n! · 1 0 n n1  Z∞ · κ 2n−α1 · GX κ; L0 ; αdκ dz. 0

(14)

Then, using the definitions of gamma function Γ· and hypergeometric function 1 F 1 · [18]: Z Γx 

0



κx−1 · e−κ dκ

κ > 0; x > 0;

(15)

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1 F 1 a; b; z 

∞ X an · zn n0

bn · n!

;

(16)

the analytical expressions of Dωpl ρ; α and Dωps ρ; α are obtained: Dωp ρ; L0 ; α 

  α Γ 1− 2 β1 α     kρ2 ηXpl α 1−α∕2 × ηXpl 1 − 1 F 1 1 − ; 1; − 2 4L    2 kρ ηX0pl α 1−α∕2 1 − 1 F 1 1 − ; 1; − − ηX0pl ; 2 4L (17) 4σ 2Rpl

where, ηX0pl  ηXpl Q0 ∕ηX0pl  Q0 , Q0  L∕kκ20 , ηXpl  L∕kκ2Xpl . σ 2Rpl is the Rytov variance for a plane wave propagating through weak non-Kolmogorov turbulence:     α πα 2 2 3−α∕2 α∕2 2 ˆ sin : L ; β1 α  4Γ − σ Rpl α  β1 AαC n π k 2 4 (18) For ease of calculation, the following approximation is adopted as [17]   α 1 − 1 F 1 1 − ; 1; −x 2      2 α∕2−2 α−4 α 2 ≈ 1− x 1 x ; (19) 2 α − 2Γα∕2 Hence, Eq. (17) can be approximately expressed as Dωp ρ;L0 ; α 

  α kρ2 Γ 2− 2 L β1 α   2 2  α∕2−2 α−4 kρ ηXpl 2 2−α∕2 1 × ηXpl α − 2Γα∕2 4L   2 2 α∕2−2  α−4 kρ ηX0pl 2 2−α∕2 1 : − ηX0pl α − 2Γα∕2 4L (20)

Dωs ρ; L0 ; α  8π 2 k2

Z

Z

L

0



0



831

1

− J 0 κρz∕LΦn1 κ; L0 ; ακdκ:

(22)

Expanding J 0 in the form of a Maclaurin series [see Eq. (13)], and considering the definitions of the Γ· function [see Eq. (15)] and generalized hypergeometric function 2 F 2 · [18]: 2 F 2 a; b; c; d; z



∞ X an bn · zn n0

cn dn · n!

;

(23)

the analytical expression for Dωs ρ; l0 ; L0 ; α is obtained:   α Γ 1− 2 β2 α     α 1 3 kρ2 ηXsp 1−α∕2 × ηXsp 1 − 2 F 2 1 − ; ; 1; ; − 2 2 2 4L    2 kρ η α 1 3 X0sp 1−α∕2 1 − 2 F 2 1 − ; ; 1; ; − − ηX0sp : 2 2 2 4L

Dωsl ρ; L0 ; α 

4σ 2Rsp

(24) Here, ηX0sp  ηXsp Q0 ∕ηX0sp  Q0 , Q0  L∕kκ 20 , and ηXsp  L∕kκ 2Xsp . σ 2Rsp is the Rytov variance for a spherical wave propagating through weak non-Kolmogorov turbulence. σ 2Rsp  β2 AαCˆ 2n π 2 k3−α∕2 Lα∕2 ;    2  α πα Γ α∕2 sin : β2 α  −4Γ 1 − 2 4 Γα

(25)

σ 2Rpl

Substituting Eq. (20) into Eq. (9), the analytical expression of variance of AOA fluctuations for a plane wave becomes   σ 2Rpl α hβ2a ipl  Γ 2− 2 kL · β1 α    2 α∕2−2 2 α−4 kD ηXpl 2 2−α∕2 × ηXpl 1 α − 2Γα∕2 4L   2 α∕2−2  2 α−4 kD ηX0pl 2 2−α∕2 1 : − ηX0pl α − 2Γα∕2 4L (21) B. Variance AOA Fluctuations for Spherical Wave under Moderate-to-Strong Non-Kolmogorov Turbulence Based on the extended Rytov theory, the wave structure function for a spherical wave propagating through moderate-to-strong non-Kolmogorov turbulence takes the form as

For ease of calculation, the following approximation is adopted as [17]   α 1 3 1 − 2 F 2 1 − ; ; 1; ; −x 2 2 2     2 α∕2−2  α−4 2−α 6 x 1 x : ≈ 6 α − 1α − 2Γα∕2

(26)

Hence, Eq. (24) can be approximately expressed as Dωs ρ; L0 ; α   σ 2Rsp α kρ2 Γ 2−  2 L 3β2 α   2 2  α∕2−2 α−4 kρ ηXsp 6 2−α∕2 1 × ηXsp α − 1α − 2Γα∕2 4L   2 2 α∕2−2  kρ η α−4 6 X0sp 2−α∕2 1 : (27) − ηX0sp α − 1α − 2Γα∕2 4L

Substituting Eq. (27) into Eq. (9), the analytical expression of variance of AOA fluctuations for spherical wave is finally derived:

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  α Γ 2− 2 3kL · β2 α    2 α∕2−2 2 α−4 kD ηXsp 6 2−α∕2 × ηXsp 1 α − 1α − 2Γα∕2 4L   2 α∕2−2  2 α−4 kD ηX0sp 6 2−α∕2 1 : − ηX0sp α − 1α − 2Γα∕2 4L

hβ2a isp 

σ 2Rsp

and turbulence outer scale on the variance of AOA fluctuations for plane and spherical waves propagating through moderate-to-strong non-Kolmogorov turbulence. The variance of AOA fluctuations as a function of Rytov variance will be plotted. The other parameters are set to: L  2000 m, D  0.05 m, and λ  1.55 μm. The turbulence outer scale’s influence on the AOA fluctuations are analyzed and shown in Figs. 1–3 under three different non-Kolmogorov turbulence cases (α  10∕3, α  11∕3, and α  3.9 are chosen). As shown, with the increase of the turbulence outer scale, the variance of AOA fluctuations increases, regardless of the variations of turbulence spectral power law value. That is because the phase fluctuations are

(28)

4. NUMERICAL CALCULATIONS In this section, calculations are performed to analyze the influences of turbulence strength, spectral power law values, −8

−8

x 10

x 10

1.2

1 L0 = 10m (this work)

1

L0 = 30m (this work) 0

L0 = 30m (this work) L = 30m (Rytov theory) 0

0.7

L0 = infinite (this work)

L0 = infinite (this work)

0.6

2 a sp

L0 = infinite (Rytov theory) pl

L0 = 10m (Rytov theory)

0.8

L = 30m (Rytov theory) 0.8

L0 = 10m (this work)

0.9

L0 = 10m (Rytov theory)

0.6 α = 10/3

L0 = infinite (Rytov theory)

0.5 α = 10/3 0.4

0.4

0.3 0.2

0.2

0.1 0 −1 10

0

10

1

2 R(pl)

σ

10

0 −1 10

2

10

0

10

1

σ2R(sp)

(a)

10

2

10

(b)

Fig. 1. Variance of AOA fluctuations derived in this study and those obtained with the Rytov theory for plane and spherical waves (α  10∕3). (a) plane wave; (b) spherical wave. −8

−8

x 10

x 10

1.4

1.4 L0 = 10m (this work)

1.2

L0 = 10m (this work)

L0 = 10m (Rytov theory)

1.2

L0 = 30m (this work) 1

L0 = 30m (this work)

L = 30m (Rytov theory)

1

0

0.6

2 a sp

L0 = infinite (Rytov theory) α = 11/3

0.8

0.6

0.4

0.4

0.2

0.2

0 −1 10

0

10

1

2 σ R(pl)

(a)

10

L = 30m (Rytov theory) 0

L0 = infinite (this work)



a pl

L0 = infinite (this work) 0.8

L0 = 10m (Rytov theory)

2

10

0 −1 10

L0 = infinite (Rytov theory) α = 11/3

0

10

1

2 R(sp)

σ

10

2

10

(b)

Fig. 2. Variance of AOA fluctuations derived in this study and those obtained with the Rytov theory for plane and spherical waves (α  11∕3). (a) Plane wave, (b) spherical wave.

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Vol. 31, No. 4 / April 2014 / J. Opt. Soc. Am. A −8

−8

x 10

x 10

7

7 L = 10m (this work)

L = 10m (this work)

0

0

L = 10m (Rytov theory)

6

L = 10m (Rytov theory)

6

0

0

L0 = 30m (this work)

L0 = 30m (this work)

L = 30m (Rytov theory)

5

L = 30m (Rytov theory)

5

0

0

L0 = infinite (this work)

L0 = infinite (this work)

L0 = infinite (Rytov theory)

2 a sp

4

2

pl

833

α = 3.9

3

4

L0 = infinite (Rytov theory)

3

α = 3.9

2

2

1

1

0 −1 10

0

10

2 σR(pl)

1

0 −1 10

2

10

10

0

10

(a)

1

2

σR(sp)

2

10

10

(b)

Fig. 3. Variance of AOA fluctuations derived in this study and those obtained with the Rytov theory for plane and spherical waves (α  3.9). (a) Plane wave, (b) spherical wave.

expressions derived in this work agree well with the results derived with the Rytov theory for both plane and spherical wave cases. When turbulence strength continues to increase, the discrepancy appears and becomes bigger and bigger for the moderate-to-strong non-Kolmogorov turbulence region. The weak perturbation theory is no longer suitable for moderate-to-strong non-Kolmogorov turbulence. It can also be seen that the variance of AOA fluctuations derived in this work always takes lower values than the case derived with the Rytov theory. This phenomenon can be explained from this point: with the increasing path length or inhomogeneity

contributed mostly by large-scale turbulence cells, when L0 is assumed with high value the wave meets a major number of large-scale turbulent cells along its propagation length, and these cells lead to higher variance of AOA fluctuations with respect to the case of lower outer scale value, where more large scales are cut out. Figures 1–3 also show the comparison curves among the variance of AOA fluctuations derived in this work and the rigorous weak non-Kolmogorov turbulence result [11] with three different α values and different turbulence outer scale values. As shown, for weak non-Kolmogorov turbulence (σ 2R ≪ 1), the

−9

−9

x 10

x 10

1.5 L = 10m

L0 = 10m

L = 100m

L0 = 100m

0

1.2

0

1

a sp

0.8





a pl

1

0.6 0.5 0.4

0.2

0

0 3

3.2

3.4

α

(a)

3.6

3.8

4

3

3.2

3.4

α

3.6

3.8

4

(b)

Fig. 4. Variance of AOA fluctuations as a function of a general spectral power law with different turbulence outer scale values under moderate-tostrong turbulence. (a) Plane wave, (b) spherical wave.

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−9

−8

x 10

x 10

8

1 L = 10m (this work) 0

7

0.9

L0 = 10m (Ref[19])

0.8

L0 = infinite (this work) L0 = infinite (Ref[19))

sp

4

2

pl

0

L = 30m (this work) 0

L = 30m (Ref[19])) 0

5

0

L = 10m (Ref[19])

L0 = 30m (this work) 6

L = 10m (this work)

α = 11/3

0.7

L = 30m (Ref[19]))

0.6

L0 = infinite (this work)

0.5

0

L0 = infinite (Ref[19])) α = 11/3

0.4

3

0.3 2 0.2 1 0 −1 10

0.1

0

10

σ2R(pl)

1

10

2

10

0 −1 10

0

10

1

σ2

10

2

10

R(sp)

(a)

(b)

Fig. 5. Variance of AOA fluctuations derived in this study and those obtained in [19] for plane and spherical waves with different turbulence outer scale values. (a) Plane wave, (b) spherical wave.

strength, the multiple scattering of an optical wave, which causes the optical wave to become increasingly less coherent as propagates through the atmospheric turbulence, weakens the turbulence effects. Different power law α and L0 values also produce different effects on the variance of AOA fluctuations. As α and L0 increase, the differences between the results derived in this work and the cases derived from the Rytov theory become less obvious. Figure 4 plots the variance of AOA fluctuations as a function of α with different turbulence outer scale values under moderate-to-strong turbulence (the Rytov variance is set to 4 in this calculation). It can be seen that, as α increases, the variance of AOA fluctuations first increases until it arrives at an optimum value, and then decreases. That is because the Aα in the effective non-Kolmogorov spectrum takes the expression that first increases with the α and then decreases with α. Figure 5 compares the results derived in this work and those in [19] for optical waves propagating through moderate-to-strong and strong Kolmogorov turbulence, in which, the results in [19] fit well with the experimental data [19]. As shown, the results derived in this work have consistency with those in [19] for the case of moderate-to-strong and strong Kolmogorov turbulence cases. The maximum differences are 1.04% (L0  10 m), 0.92% (L0  30 m), and 0.75% (L0  ∞) for the plane wave case. for the spherical wave, the maximum differences are 7.36% (L0  10 m), 6.64% (L0  30 m), and 5.52% (L0  ∞), respectively. The Kolmogorov turbulence is one special case of non-Kolmogorov turbulence and this comparison proves experimentally, to some extent, the effectiveness of our derivations.

5. CONCLUSION In this study, with the generalized von Kármán spectrum and the extended Rytov theory method, new theoretical expressions of the variance of AOA fluctuations are derived for

optical waves propagating through moderate-to-strong nonKolmogorov turbulence with horizontal path. Calculation results show that the presence of a finite turbulence outer scale decreases the variance of AOA fluctuations obviously compared with those in [13]. Also, the multiple scattering effect of an optical wave makes the results derived in this work always take a lower value than those obtained with the Rytov theory under moderate-to-strong (or strong) non-Kolmogorov turbulence case. In addition, the expressions in this work have good consistency with the experimental data for the case of moderate-to-strong (or strong) Kolmogorov turbulence (α  11∕3). Further experimental verification needs to be performed in future. Despite this, this work should be considered the first step of a more complete analysis, based on new experimental investigations.

ACKNOWLEDGMENTS This work is partly supported by the Beijing Natural Science Foundation (4122047), and the Innovation Foundation of AVIC (No. CXY2010BH02).

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Angle of arrival fluctuations considering turbulence outer scale for optical waves' propagation through moderate-to-strong non-Kolmogorov turbulence.

Based on the generalized von Kármán spectrum and the extended Rytov theory, new analytic expressions for the variance of angle of arrival (AOA) fluctu...
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