Refractive-index fluctuations spectrum considering the general distribution of turbulence cells in moderate-to-strong anisotropic turbulence

Optik 154 (2018) 473–484

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Optik journal homepage: www.elsevier.de/ijleo

Original research article

Refractive-index fluctuations spectrum considering the general distribution of turbulence cells in moderate-to-strong anisotropic turbulence Linyan Cui School of Astronautics, Beihang University, Beijing 100191, China

a r t i c l e

i n f o

Article history: Received 24 July 2017 Accepted 18 October 2017 Keywords: Anisotropic turbulence Refractive-index fluctuations Turbulence spectrum

a b s t r a c t For the moderate-to-strong anisotropic turbulence, the previously derived turbulence refractive-index fluctuations spectral models assumed the circular symmetric distribution of turbulence cells in the plane orthogonal to the direction of propagation. For the real anisotropic turbulence, this kind of distribution is very special and not always fit the real turbulence. In this work, to describe the general distribution of turbulence cells in the moderate-to-strong anisotropic turbulence, new turbulence refractive-index fluctuations spectral model is derived with the extended Rytov approximation theory. Then, it is applied to investigate the irradiance scintillation index for optical plane and spherical waves propagating through moderate-to-strong anisotropic turbulence. Two anisotropic factors are introduced to describe the asymmetric distribution of turbulence cells. Compared with the Kolmogorov turbulence, the general spectral power law in the range of 3–4 is also considered. The derived irradiance scintillation index models of optical waves are applicable in a wide range of turbulence strength. In the special case of weak anisotropic turbulence, they have good consistency with those derived in weak anisotropic turbulence. © 2017 Elsevier GmbH. All rights reserved.

1. Introduction In recent years, the investigations of anisotropic turbulence have gained more and more interests as new experiments and theoretical developments become available. They have shown that the atmosphere turbulence exhibits anisotropic properties [1–15] at high altitudes, such as in the stratosphere, as well as on the order of meters above the ground [16–19]. For the anisotropic turbulence, the horizontal size of the turbulence outer scale cells is typically tens of meters across or, in some cases, kilometers across. While the vertical size of the turbulence outer scale cells is usually confined to a few meters. According to the Richardson cascade theory, the sizes of anisotropic turbulence cells in horizontal direction will be bigger than the vertical ones. The asymmetric distribution of turbulence cells will produce non-negligible effects on the optical waves’ propagation. Recently, the investigations are mainly focused on the weak (characterized byR2 << 1, where R2 is the Rytov variance) and moderate-to-strong anisotropic turbulence. With the Rytov approximation theory, a series of turbulence effects models have been derived for plane/spherical/Gaussian beam waves propagating through weak anisotropic turbulence [12–15,20,21]. For the moderate-to-strong anisotropic turbulence, the modified anisotropic turbulence refractive-index fluctuations spectral models [22] have been derived with the extended Rytov approximation theory. They modified the conventional anisotropic turbulence refractive-index fluctuations spectral models [11] with a filter

E-mail address: [email protected] https://doi.org/10.1016/j.ijleo.2017.10.082 0030-4026/© 2017 Elsevier GmbH. All rights reserved.

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L. Cui / Optik 154 (2018) 473–484

function. Based on these spectral models and the extended Rytov approximation theory, the irradiance scintillation index and the modulation transfer functions have been derived for plane and spherical waves propagating through moderate-tostrong anisotropic turbulence [22,23]. Still they assumed the circular symmetric distribution of turbulence cells in the plane orthogonal to the direction of propagation. This special assumption may not fit the real anisotropic turbulence. The more general distribution of turbulence cells in the anisotropic turbulence should be considered for accurate modeling. In this work, new atmosphere turbulence refractive-index fluctuations spectral model will be derived for optical plane and spherical waves propagating through moderate-to-strong anisotropic turbulence. It can characterize the asymmetric turbulence cells in the plane orthogonal to the direction of propagation. Based on this, the irradiance scintillation index models of plane and spherical waves in moderate-to-strong anisotropic turbulence will also be developed. Last, calculations will be performed to analyze the derived turbulence refractive-index fluctuations spectrum and the irradiance scintillation index models. 2. Refractive-index fluctuations spectrum considering the general distribution of turbulence cells in moderate-to-strong anisotropic turbulence The anisotropic turbulence refractive-index fluctuations spectrum which characterizes the general distribution of turbulence cells in weak anisotropic turbulence has been derived previously [22]:





 −˛

 ˚n , ˛, ux , uy = A (˛) · Cˆ n2 · ux uy · 



= A (˛) · Cˆ n2 · ux uy · z2 + u2x x2 + u2y y2

(1)

− ˛

2.

Note that this spectral model is only valid in the inertial subrange (1/L0 «  « 1/l0 ). In which,  is the wavenumber related to the turbulence cell size.  = x2 + y2 + z2 . x , y , and z are the components of  in the x, y, and z directions. l0 is the turbulence inner scale with the unit of millimeter, and L0 is the turbulence outer scale with the unit of meter. In the theoretical investigations of optical waves propagating through weak anisotropic turbulence, Eq. (1) is extended to the whole range (0 <  < ∞) by assuming l0 → 0 and L0 → ∞. ␣ is the general spectral power law value in the range 3–4. A (˛) is a constant which maintains consistency between the refractive index structure function and its power spectrum. A (˛) =



1 ˛  (˛ − 1) cos 2 42



(2)





 (·) is the gamma function. When ␣ = 11/3, A 11/3 = 0.033, the Kolmogorov turbulence is exhibited. Cˆ n2 = ˇCn2 is the generalized structure parameter with unit [m3−˛ ],ˇ is a dimensional constant with unit [m11/3−˛ ], and Cn2

is the structure parameter for the Kolmogorov turbulence with unit of m−2/3 . Compared with the spectrum derived in [11], Eq. (1) introduced two anisotropic factors of ux and uy to parameterize the general distribution of asymmetric turbulence cells in horizontal and vertical directions. When ux = uy , it exhibits the circular symmetric distribution of turbulence cells in the plane orthogonal to the direction of propagation. Furthermore, when ux = uy = 1, the sizes of turbulence cells in horizontal and vertical directions are the same, and the isotropic turbulence is shown. At this time, Eq. (1) reduces to the conventional isotropic turbulence spectrum.   In the derivation of Eq. (1),  = z2 + u2x x2 + u2y y2 is introduced to make the anisotropic turbulence refractive-index 





fluctuations spectrum ˚n (·) isotropic in the stretched wave number space of x = ux x , y = uy y , z = z . By invoking the Markov approximation, which assumes that the index of refraction is delta-correlated at any pair of points located along the direction of propagation, the component of z in Eq. (1) can be ignored. Hence, Eq. (1) becomes [22]:







˚n , ˛, ux , uy = A (˛) · Cˆ n2 · ux uy · u2x x2 + u2y y2

− ˛2

.

(3)

For the moderate-to-strong anisotropic turbulence, according to the extended Rytov approximation theory, the modified anisotropic turbulence refractive-index fluctuations spectral model can be expressed as:







 







˚n1 , ˛, ux , uy = ˚n , ˛, ux , uy G , ˛, ux , uy









(4)

G , ˛, ux , uy = GX , ˛, ux , uy + GY , ˛, ux , uy ,



(5)







G , ˛, ux , uy is the amplitude spatial filter function, and it is composed by the large-scale filter GX , ˛, ux , uy and the





small-scale filter GY , ˛, ux , uy . In this study, the finite turbulence inner and outer scales are not considered.







GX , ˛, ux , uy = exp −

X2



2 ˛, ux , uy



   , GY , ˛, ux , uy = 

 2

˛

2 + Y ˛, ux , uy

˛/2 .

(6)

L. Cui / Optik 154 (2018) 473–484



475



In which, X ˛, ux , uy is a large-scale (or refractive) cutoff spatial frequency much like an inner-scale parameter, and





Y ˛, ux , uy is a small-scale (or diffractive) cutoff spatial frequency similar to an outer-scale parameter.



1

X2 ˛, ux , uy



2

L L   = c1 + c2 k k0 ˛, ux , uy





, 2Y ˛, ux , uy = c3

k 1  . + c4 · L 02 ˛, ux , uy

(7)

L is the propagation path length. k = 2/ is the wavenumber of optical wave, and  is  the optical  wavelength. c1 , c2 , c3 , and c4 are the undetermined parameters which need to be calculated in this work. 0 ˛, ux , uy is the coherence radius



of optical wave in weak anisotropic turbulence. G , ˛, ux , uy







acts as an irradiance spatial filter function that permits





only low-pass spatial frequencies < X ˛, ux , uy or high-pass spatial frequencies > Y ˛, ux , uy to influence the final irradiance scintillation index of an optical wave.

3. Irradiance scintillation index for plane and spherical waves considering the general distribution of turbulence cells in anisotropic turbulence According to the extended Rytov approximation theory, the modified anisotropic turbulence refractive-index fluctuations spectral model can be applied to investigate the irradiance scintillation index models of optical waves in moderate-to-strong anisotropic turbulence. Within the extended Rytov approximation framework, in the special conditions of very weak (R2 << 1) and very strong (R2 >> 1) turbulence, the derived irradiance scintillation index of optical waves in the moderate-to-strong turbulence should have good consistency with the results derived in weak and strong anisotropic turbulence. With these two conditions, the undetermined parameters of c1 , c2 , c3 , and c4 in the modified anisotropic turbulence refractive-index fluctuations spectral model will be obtained. Meanwhile, the irradiance scintillation index models of plane and spherical waves in moderate-to-strong anisotropic turbulence will be finally determined. This section is divided into three subsections, and they are organized as follows: first, based on the Rytov approximation theory and the anisotropic turbulence refractive-index fluctuations spectral model (Eq. (1)), the irradiance scintillation index models of optical plane and spherical waves in weak anisotropic turbulence will be introduced. Second, in view of the classical relation between the turbulence refractive-index fluctuations spectral model and the irradiance scintillation index model of an optical wave in saturation turbulence region, the irradiance scintillation index models of optical plane and spherical waves in strong anisotropic turbulence will be derived. Third, based on the extended Rytov approximation theory and the modified anisotropic turbulence refractive-index fluctuations spectral model (Eq. (4)), the irradiance scintillation index models of optical plane and spherical waves in moderate-to-strong anisotropic turbulence will be developed. The general distribution of turbulence cells in the anisotropic turbulence is considered in the irradiance scintillation index models developed in these three subsections. Last, in view of the relations of the irradiance scintillation index models derived in weak, moderate-tostrong, and strong turbulence in the special cases of R2 << 1 and R2 >> 1, the undetermined parameters of c1 , c2 , c3 , and c4 in the modified anisotropic turbulence refractive-index fluctuations spectral model will be obtained. At this time, both the modified anisotropic turbulence refractive-index fluctuations spectrum and the irradiance scintillation index model of optical waves in moderate-to-strong anisotropic will be finally acquired.

3.1. Irradiance scintillation index of optical waves considering the general distribution of turbulence cells in weak anisotropic turbulence For weak anisotropic turbulence, based on the Rytov approximation theory, the irradiance scintillation index for plane and spherical waves can be expressed as

2 R(pl)





L ∞ 2 2

˛, ux , uy = 8 k

0

2 R(sp)





L ∞ 2 2





˚n , ˛, ux , uy 0





1 − cos

2 z k

 ddz,

(8)

0

˛, ux , uy = 8 k





˚n , ˛, ux , uy



1 − cos



2 z 1 − z/L k

 

ddz

(9)

0





2 2 where R(pl) ˛, ux , uy and R(sp) ˛, ux , uy represent the irradiance scintillation indexes (also called Rytov variance) for

plane and spherical waves in weak anisotropic turbulence, respectively.

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L. Cui / Optik 154 (2018) 473–484





For the derivation convenience, it needs to change the stretched coordinate system for the spectrum ˚n , ˛, ux , uy to an isotropic one with the following substitutions x =

qy qx q cos q sin = , y = = ,q = x x y y

dx dy =

 q2x + q2y .

dqx dqy qdqd = . x y x y





(10)

− ˛ 2



ˆ (˛) · Cˆ n2 · x y · 2x x2 + 2y y ˚n , ˛, x , y = A

2

ˆ (˛) · Cˆ n2 .x y · q−˛ . =A

 Then, making the variable substitution of q1 = q

cos2 2 x

+

sin2 and 2 y

using the definition of gamma function [24]:

∞ x−1 · e− d,

 (x) =

(11)

0

Eqs. (8) and (9) become



2 R(pl) ˛, ux , uy

ˇ1 = 4 · 

2 R(sp)





1 = ˇ1 · A (˛) Cˆ n2 2 k3−˛/2 L˛/2 2

 ˛ −

2

· sin



˛, ux , uy =

ˇ2 = −4 · 



1−

 ˛  4

2

· sin

d

cos2 2x

+

sin2

˛ − 2 2

2y

,

(12)

0

,

ˇ2 · A (˛) Cˆ n2 2 k3−˛/2 L˛/2

 ˛

2

 ˛   2 ˛/2 ·

4

1 2

 (˛)

2 d

cos2 2x

+

sin2 2y

˛ − 2 2

,

0

(13)

.

3.2. Irradiance scintillation index of optical waves considering the general distribution of turbulence cells in strong anisotropic turbulence In the saturation region (R2 (˛) >> 1), the irradiance scintillation index is given by [24]:

1 ∞





⎧ 1 ⎨ 

 L

I2 S ˛, ux , uy = 1 + 82 L3 ·

exp

−

Ds

⎩

k

⎫0 ⎬

 w (, ε) d





5 ˚n , ˛, ux , uy · w2 (ε, ε)

⎭

0

(14)

ddε,

0

   ˜  1 − ε , <ε   . w (, ε) = ˜ , ε 1 − 

(15)

>ε

˜ = 0 for plane wave, and ˜ = 1for spherical wave. Ds [·]denotes the phase structure function and takes the form as here



⎡



Ds , ˛, ux , uy = 42 A (˛) Cˆ n2 k2 L · ⎣−

1 2

2 d 0

cos2 2x



2

+

sin 2y







 ˛/2

·

2 4

˛ − 1 2

⎤ ⎦ (16)

˛ − 2 2



 1 − ˛/2

.

L. Cui / Optik 154 (2018) 473–484



477



Substituting Eq. (16) intoI2 S ˛, ux , uy , then making use of Eq. (10) for the coordinate system transformation and adopt-



cos2 2 x

ing the variable substitution of q1 = q

+

sin2 , 2 y

the derivations for the irradiance scintillation index of optical waves

in strong anisotropic turbulence will be converted to the isotropic turbulence case. Last, using the definitions of gamma function (see Eq. (11)) and gauss hypergeometric function 2 F 1 (A, B; C; Z) [24]  (C) 2 F 1 (A, B; C; Z) =  (B) ·  (C − B)

1 t B−1 · (1 − t)C−B−1 · (1 − tZ)−A dt,

(17)

0

for the integrations of dq1 and dε appearedin the derivations of irradiance scintillation index of optical waves in strong  anisotropic turbulence, the expressions of I2 S ˛, ux , uy for plane and spherical waves in strong anisotropic turbulence are obtained.





4(˛−4)





4(˛−4)





˛−2 I2 S(pl) ˛, ux , uy = 1 + r1 · I1 · R(pl) ˛−2 I2 S(sp) ˛, ux , uy = 1 + r2 · I2 · R(sp)







˛, ux , uy ,

 

(18)



˛, ux , uy .

(19)

whereI2 S(pl) ˛, ux , uy andI2 S(sp) ˛, ux , uy represent separately the irradiance scintillation indexes of plane and spherical waves in strong anisotropic turbulence. r1 , I1 ,r2 , and I2 take the forms as



1 r1 = · [2] ˛−2 I1 =

1 2F 1 ˛−3

  (3 − ˛) (˛ − 10)  1 − ˛/2 ˛−2   · −  ˛/2

6 − ˛ ˛−2

, ˛ − 3; ˛ − 2;

 ˛−2 ˛−1

˛ − 6 ˛−2

·

8 − 2˛ · ˇ1 (˛) ˛ − 2 ,

6 − ˛  ˛−2



(20)

,



  (3 − ˛) (˛ − 10)  1 − ˛/2 1 ˛−2   · − · [2] r2 = ˛−2  ˛/2

˛ − 6 ˛−2

·

8 − 2˛  · ˇ2 (˛) ˛ − 2 ,

6 − ˛  ˛−2

(21)

6−˛  2 (˛ − 3) ˛ . I2 = (˛ − 1) − 2 ·  (2˛ − 6) 3.3. Irradiance scintillation index of optical waves considering the general distribution of turbulence cells in moderate-to-strong anisotropic turbulence For moderate-to-strong atmosphere turbulence, according to the extended Rytov approximation theory, the received optical wave irradiance at the receiver can be modeled as a modulation process. Specifically, the small-scale (diffractive) fluctuations of optical wave irradiance are multiplicatively modulated by statistically independent large-scale (refractive) fluctuations of optical wave irradiance. At this time, the irradiance scintillation index of optical waves in moderate-to-strong turbulence takes the following form as [24]:









2 2 2 I2 = exp ln − 1 = exp ln + ln − 1. I X Y

(22)

2 and 2 represent the large-scale and small-scale log-irradiance scintillation index of optical wave, respectively. hereln X ln Y For moderate-to-strong anisotropic turbulence, the general spectral power law and the general distribution of asymmetric turbulence cells in anisotropic turbulence need to be considered. According to the extended Rytov approximation theory, the irradiance scintillation index of plane wave in moderate-to-strong anisotropic turbulence takes the form as:













2 2 2 I(pl) ˛, ux , uy = exp ln ˛, ux , uy + ln ˛, ux , uy X(pl) Y (pl)

2 ln X(pl)





L ∞ 2 2

˛, ux , uy = 8 k





− 1,

















˚n , ˛, ux , uy GX , ˛, ux , uy 0

2 ln Y (pl)





1 − cos

2 z k

 ddz,

(24)

ddz.

(25)

0

 

L ∞

2 2

˛, ux , uy = 8 k

˚n , ˛, ux , uy GY , ˛, ux , uy 0

(23)

1 − cos

2 z k



0

The detailed derivations for the irradiance scintillation index of plane wave in moderate-to-strong anisotropic turbulence are given as follows. First, adopting Eq. (10) for the coordinate system transformation just like it was did in subsections of

478

L. Cui / Optik 154 (2018) 473–484

 3.1 and 3.2 to simplify the mathematical derivations. Second, making the variable substitutions of q1 = q

=

Lq2 1 k

cos2 2 x

+

sin2 2 y

and

to further simplify the derivations.

Third, adopting the geometrical optics approximation of 1 − cos x ≈

x2

for the large-scale log-irradiance scintil2   2 lation index ln ˛, u , u and the approximation of 1 − cosx ≈ 1 for the small-scale log-irradiance scintillation x y  X(pl)  2     2 2 indexln Y (pl) ˛, ux , uy , ln X(pl) ˛, ux , uy and ln Y (pl) ˛, ux , uy can be finally expressed as: 2 ln X(pl)







˛, ux , uy =





2 ˛, ux , uy = ln Y (pl)





2 3 − ˛/2

3− ˛









2 2 · X(pl) ˛, ux , uy · R(pl) ˛, ux , uy .

3ˇ1

(26)

  2   1− ˛ 8 2 · Y (pl) ˛, ux , uy · R(pl) ˛, ux , uy . − 2) ˇ (˛ 1 



(27)



in which, X(pl) ˛, ux , uy and Y (pl) ˛, ux , uy represent the ratios of the Fresnel zone area to the turbulence cell area:







2 ˛, ux , uy LX(pl)

X(pl) ˛, ux , uy =



LY2 (pl)

Y (pl) ˛, ux , uy =



˛, ux , uy

1

=

k









k

L



c1(pl) + c2(pl) = c3(pl) + c4(pl)



k02 (pl) ˛, ux , uy L



k02 (pl) ˛, ux , uy



, (28)

.

where0(pl) ˛, ux , uy is the coherence radius for plane wave in weak anisotropic turbulence:

⎡

  1 1 0(pl) ˛, ux , uy = [1 (˛)] ˛−2 · ⎣2 A (˛) Cˆ n2 k2 L 2 

1 (˛) = −



2 d

cos2 2x



2

+

sin 2y

 ˛−2 2

−1 ⎤ ˛−2

⎦

,

(29)

0

 

 ˛/2

 1 − ˛/2 · 23−˛

.

(30)

Last, the undetermined parameters of c1(pl) , c2(pl) , c3(pl) , and c4(pl) for plane wave will be derived with the extended Rytov









2 2 << 1 and R(pl) >> 1, X(pl) ˛, ux , uy and Y (pl) ˛, ux , uy in Eq. approximation theory. In the two special cases of R(pl)

(28) can be approximately expressed as





X(pl) ˛, ux , uy ≈









1 c1(pl)

Y (pl) ˛, ux , uy ≈ c3(pl)

X(pl) ˛, ux , uy ≈





2 R(pl) << 1 .



2 ˛, ux , uy k · 0(pl)

 ,

L · c4(pl)



2 k · 0(pl) ˛, ux , uy

(31)



L · c2(pl)



Y (pl) ˛, ux , uy ≈

 ,





2 R(pl) >> 1 .

(32)

In addition, according to the extended Rytov approximation theory, in these two special cases of R2 << 1 and R2 >> 1, the derived irradiance scintillation index model of optical wave in moderate-to-strong anisotropic turbulence should has good consistency with those derived in weak and strong anisotropic turbulence. That is, the following two relations will be satisfied.









2 2 exp ln ˛, ux , uy + ln ˛, ux , uy X(pl) Y (pl)

≈

2 ln X(pl)

≈

2 R(pl)





˛, ux , uy

˛, ux , uy









2 + ln Y (pl)





˛, ux , uy





2 2 ˛, ux , uy + ln ˛, ux , uy exp ln X(pl) Y (pl)



≈ I2 S(pl) ˛, ux , uy





−1

 ,



−1

 ,



2 R(pl) << 1

(33)



2 R(pl) >> 1

(34)

L. Cui / Optik 154 (2018) 473–484







Substituting X(pl) ˛, ux , uy and Y (pl) ˛, ux , uy











479

which appear in Eqs. (31) and (32) into Eqs. (26) and (27),

2 2 ln ˛, ux , uy and ln ˛, ux , uy in the two special cases of R2 << 1 and R2 >> 1will be obtained. Then, putting X(pl) Y (pl)

them into Eqs. (33) and (34), the expressions for c1(pl) ,c2(pl) , c3(pl) and c4(pl) are obtained:

c1(pl) =

c3(pl)



2 3 − ˛/2



2 6−˛

3ˇ1 × 0.49



3ˇ1 · r1 · I1



, c 2(pl) =

2 

˛−6

4 3 − ˛/2

2   · 1 (˛) · ˇ1 ˛ − 2 , (35)

 2  − 2) · ln 2  2 0.51 × (˛ − 2) · ˇ1 2 − ˛ (˛ 2−˛. , c 4(pl) = = 8 8 · 1 (˛)

With the derived c1(pl) ,c2(pl) , c3(pl) and c4(pl) , both the modified anisotropic turbulence refractive-index fluctuations spectral model and the irradiance scintillation index model for plane wave propagating through moderate-to-strong anisotropic turbulence are acquired. The general distribution of asymmetric turbulence cells in anisotropic turbulence is considered.

⎡

2 I(pl)



˛, ux , uy





⎢ ⎢ = exp ⎢ ⎣

2 0.49R(pl) ˛, ux , uy 4 ˛−2





1 + fX(pl) · R(pl) ˛, ux , uy







X(pl) ˛, ux , uy =



3ˇ1 × 0.49



3− ˛2 +

4 ˛−2





1 + fY (pl) · R(pl) ˛, ux , uy

⎥ ⎥ − 1.  ˛2 −1 ⎥ ⎦

(36)

2 

6−˛

⎤,

4   ⎣1 + fX(pl) ·  ˛ − 2 ˛, ux , uy ⎦ R(pl)





2 3 − ˛/2

⎡





2 ˛, ux , uy 0.51R(pl)

⎤

Y (pl) ˛, ux , uy =

⎡

0.51 × (˛ − 2) · ˇ1 8

 2

2−˛

(37)

·

⎤

4   ⎣1 + fY (pl) ·  ˛ − 2 ˛, ux , uy ⎦ , R(pl)

fX(pl)

 r ·I  2 1 1 ˛−6 = 2 × 0.49

, fY (pl) =

 ln 2  2

2−˛ .

0.51



2 For the spherical wave, the irradiance scintillation index I(sp) ˛, ux , uy



in moderate-to-strong anisotropic turbu-



2 lence can also be expressed by the large-scale and small-scale log-irradiance scintillation indexes of ln ˛, ux , uy X(sp)









2 and ln ˛, ux , uy . Y (sp)









2 2 2 I(sp) ˛, ux , uy = exp ln ˛, ux , uy + ln ˛, ux , uy X(sp) Y (sp)



L ∞



2 ˛, ux , uy = 82 k2 ln X(sp)





· 1 − cos

2 ln Y (sp)





2 z 1 − z/L

· 1 − cos

0  





k

 (39)

ddz,

82 k2

2 z 1 − z/L



(38)

0

L ∞





− 1,

 · ˚n , ˛, ux , uy · GX , ˛, ux , uy

k

˛, ux , uy =









0  







 · ˚n , ˛, ux , uy · GY , ˛, ux , uy 0

ddz.

 (40)

480

L. Cui / Optik 154 (2018) 473–484

Following the same steps as those adopted to derive scintillation index of plane wave in moderate-to the irradiance  strong anisotropic turbulence, the parameters of 0(sp) ˛, ux , uy (coherence radius for spherical wave in weak anisotropic turbulence),c1(sp) , c2(sp) , c3(sp) , and c4(sp) for spherical wave will be obtained:

⎡

 2 2 1   cos 1 sin2 ⎢ 2 2 2 ˆ ˛ − 2 · ⎣ A (˛) Cn k L 0(sp) ˛, ux , uy = [2 (˛)] d + 2 2x 2y 

2 (˛) = −

c1(sp) =



c3(sp) =

⎦

, (41)

0

 ˛/2 (˛ − 1)









 1 − ˛/2 · 23−˛

2 3 − ˛/2

.

2 6−˛

30ˇ2 × 0.49



⎤ −1 ˛−2 ˛−2 2 ⎥

0.51 × (˛ − 2) · ˇ2 8



30ˇ2 · r2 (˛) · I2 (˛)



, c 2(sp) =



˛−6

4 3 − ˛/2

 2

2−˛

2

, c 4(sp) =



2   · 2 (˛) · ˇ2 ˛ − 2 , (42)

 2 − 2) · ln 2

(˛ 8 · 2 (˛)

2−˛.

With the derivedc1(sp) , c2(sp) , c3(sp) , and c4(sp) , the modified anisotropic turbulence refractive-index fluctuations spectral model and the irradiance scintillation index model for spherical wave in moderate-to-strong anisotropic turbulence are finally acquired:

⎡

⎤

⎢ ⎥ ⎢ ⎥ ⎢ ⎥     ⎢ ⎥ 2 2 0.49R(sp) ˛, ux , uy 0.51R(sp) ˛, ux , uy   ⎢ ⎥ 2 I(sp) ˛, ux , uy = exp ⎢ + ⎥ − 1. (43) ˛ ˛ ⎢⎡ ⎥ ⎡ ⎤ ⎤ 3− − 1 4 4 ⎢ ⎥ 2 2 ⎢ ⎥     ˛ − 2 ⎣ ⎣1 + fX(sp) ·  ˛ − 2 ˛, ux , uy ⎦ ⎦ ⎣ ⎦ 1 + fY (sp) · R(sp) ˛, ux , uy R(sp) in which, the large-scale and small-scale log-irradiance scintillation indexes of spherical wave in moderate-to-strong anisotropic turbulence take the expressions as: 2 ln X(sp)







˛, ux , uy =





2 ln ˛, ux , uy = Y (sp)





2 3 − ˛/2 30ˇ2



3− ˛









2 2 · X(sp) ˛, ux , uy · R(sp) ˛, ux , uy ,

(44)

  2   1− ˛ 8 2 · Y (sp) ˛, ux , uy · R(sp) ˛, ux , uy . (˛ − 2) ˇ2 

(45)



where X(sp) ˛, ux , uy and Y (sp) ˛, ux , uy take the forms as:





X(sp) ˛, ux , uy =





Y (sp) ˛, ux , uy =

f X(sp) (˛) =



2 LX(sp) ˛, ux , uy



LY2 (sp) ˛, ux , uy k

 r ·I  2 2 2 ˛−6 2 × 0.49





6−˛

2 3 − ˛/2

=

k

30ˇ2 × 0.49





2

1 + fX(sp) ·



2 R(sp)





2 ˛−2

,

˛, ux , uy ⎡ ⎤ 4  2   0.51 (˛ − 2) · ˇ2 2 − ˛ ˛ − 2 ˛, u , u ⎦ , · ⎣1 + fY (sp) · R(sp) = x y 8

, fY (sp) (˛) =

(46)

 ln 2  2

2−˛ .

0.51

The derived irradiance scintillation index models of optical plane and spherical waves in moderate-to-strong anisotropic turbulence consider the general distribution of turbulence cells in the anisotropic turbulence. Compared with previously derived models in [22] which assumed the circular symmetric distribution of turbulence cells in the plane orthogonal to the direction of propagation, the results derived in this work should fit better with the real anisotropic turbulence.

L. Cui / Optik 154 (2018) 473–484

481

Fig. 1. Key parameters of X(pl) (˛, ux , uy ) and Y (pl) (˛, ux , uy )in the modified anisotropic turbulence refractive-index fluctuations spectral model for plane wave as a function of turbulence strength with different anisotropic factors and ␣values. (a): ␣ = 3.1; (b): ␣ = 10/3; (c): ␣ = 11/3.

In addition, from Eqs. (36) and (43), it can be see that in the special case of weak anisotropic turbulence, they approximately become





















2 2 I(pl) ˛, ux , uy ≈ exp R(pl) ˛, ux , uy

2 2 I(sp) ˛, ux , uy ≈ exp R(sp) ˛, ux , uy









2 − 1 ≈ R(pl) ˛, ux , uy

2 − 1 ≈ R(sp) ˛, ux , uy

(47) (48)

At this time, the derived irradiance scintillation index models of optical waves in moderate-to-strong anisotropic turbulence reduce to the results derived in weak anisotropic turbulence. 4. Calculations and analyses In this section, numerical calculations are carried out to analyze the influences of anisotropic turbulence on the derived modified anisotropic turbulence refractive-index fluctuations spectral models and the irradiance scintillation index models for optical plane and spherical waves propagating through moderate-to-strong anisotropic turbulence. turbulence refractive-index fluctuations spectral models, the cut-off spatial frequenFor themodified anisotropic      cies of X ˛, ux , uy andY ˛, ux , uy are the key parameters. They are related to X ˛, ux , uy and Y ˛, ux , uy with   L2 (˛,ux ,uy )   L2 (˛,ux ,uy )     and Y ˛, ux , uy = Y k . From the definitions of X ˛, ux , uy and X ˛, ux , uy , the

X ˛, ux , uy = X k



relations between the Fresnel size (L/k) and the turbulence cell size can be represented directly. Therefore, X ˛, ux , uy







and X ˛, ux , uy as a function of turbulence strength (here, the Rytov variance is adopted) will be analyzed just like the previous  researches  did in  [22,24].

X ˛, ux , uy and X ˛, ux , uy as a function of turbulence strength for both plane and spherical waves are plotted in Figs. 1 and 2. Different anisotropic factors and general spectral power law values are chosenas examples  only for theoretical   analysis. As shown, when one of the two anisotropic factors is fixed (here, ux = 1), X(pl) ˛, ux , uy and X(sp) ˛, ux , uy become bigger with the increased uy value. In this case, the turbulence cell that acts as the large-scale (or refractive) cutfrequency will become smaller and smaller than the Fresnel size. In contrast, the opposite trend appears for off spatial    

Y (pl) ˛, ux , uy and Y (sp) ˛, ux , uy , which means the turbulence cell that acts as the small-scale (or diffractive) cutoff



spatial frequency will become bigger and bigger than the Fresnel size. At this time, the amplitude spatial filter G , ˛, ux , uy











will permit more turbulence cells with  < X ˛, ux , uy or  > Y ˛, ux , uy to influence the optical waves’ propagation









through anisotropic turbulence. In this part, the maximum percentage differences of X ˛, ux , uy and Y ˛, ux , uy between the anisotropic turbulence and the isotropic turbulence are calculated quantitatively and listed in Table 1. For comparison purpose, the maximum percentage differences of X (˛, ς ) and Y (˛, ς ) between the anisotropic turbulence and the isotropic turbulence derived in [22] are also calculated and listed in Table 2. In which, the circular symmetric distribution of turbulence

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Fig. 2. Key parameters of X(sp) (˛, ux , uy ) and Y (sp) (˛, ux , uy )in the modified anisotropic turbulence refractive-index fluctuations spectral model for spherical wave as a function of turbulence strength with different anisotropic factors and ␣values. (a): ␣ = 3.1; (b): ␣ = 10/3; (c): ␣ = 11/3. Table 1 The maximum percentage difference of X (˛, ux , uy ) and Y (˛, ux , uy ) between the anisotropic and isotropic turbulence cases derived in this work. ˛ = 3.1, ux = 1

X(pl) (˛, ux , uy )

Y (pl) (˛, ux , uy )

X(sp) (˛, ux , uy )

Y (sp) (˛, ux , uy )

˛ = 10/3, ux = 1

˛ = 11/3, ux = 1

uy = 2

uy = 3

uy = 4

uy = 2

uy = 3

uy = 4

uy = 2

uy = 3

uy = 4

67.19% 40.23% 66.92% 40.23%

96.46% 49.15% 96.01% 49.15%

111.05% 52.67% 110.49% 52.67%

65.09% 39.45% 64.79% 39.45%

91.50% 47.81% 91.00% 47.81%

103.96% 51.00% 103.36% 51.00%

62.22% 38.227% 61.86% 38.27%

85.03% 45.85% 84.47% 45.85%

95.05% 48.62% 94.39% 48.62%

Table 2 The maximum percentage difference of X (˛, ς ) and Y (˛, ς )between the anisotropic and isotropic turbulence cases derived in [22]. ˛ = 3.1

X(pl) (˛, ς )

Y (pl) (˛, ς )

X(sp) (˛, ς )

Y (sp) (˛, ς )

˛ = 10/3

˛ = 11/3

ς=2

ς=3

ς=4

ς=2

ς=3

ς=4

ς=2

ς=3

ς=4

298.32% 74.97% 295.50% 74.97%

789.99% 88.85% 773.51% 84.85%

1.467e3% 93.71% 1.414e3% 93.71%

297.29% 74.88% 293.97% 74.88%

783.92% 88.74% 764.69% 88.74%

1.447e3% 93.59% 1.386e3% 93.60%

259.63% 74.55% 291.51% 74.56%

774.63% 88.36% 750.81% 88.36%

1.416e3% 93.19% 1.344e3% 93.19%

cells in the plane orthogonal to the direction of propagation was assumed in [22], only an anisotropic factor of ς is applied to describe the asymmetric turbulence cells in anisotropic turbulence. From Tables 1 and 2, it can be seen that the variable uy produces less effect on the derived modified anisotropic turbulence refractive-index fluctuations spectral models compared with the results shown in [22]. Both of these two anisotropic factors of ux and uy are indispensable in the analysis of optical waves’ propagation through anisotropic turbulence. They produce joint impact on the final results. Then, the influences of general spectral power law and anisotropic factors on the irradiance scintillation index models derived in this work are analyzed in detail. The irradiance scintillation index models of optical plane and spherical waves as a function of turbulence strength with different general spectral power law and anisotropic factor values are plotted in Fig. 3. Still, one of the two anisotropic factors is fixed, and the other one varies. Due to the lack of sufficient experiments for the anisotropic turbulence, all the parameters in the calculations are chosen as examples only for theoretical analysis. From Fig. 3, it can be seen that the irradiance scintillation becomes severer with the increased turbulence strength until arrives at an optimum value where the large-scale inhomogeneities achieves its strongest effect. The multiple scattering effects introduced by the strong turbulence weaken the focusing effect of turbulence cells, which subsequently decrease the irradiance scintillation index and make it saturate at a level. The physical explanations for this phenomenon are the same as those stated in our previous work [22]. From this figure, it can also be seen that with the increase of uy , the saturation

L. Cui / Optik 154 (2018) 473–484

483

Fig. 3. Irradiance scintillation index of optical plane and spherical waves as a function of turbulence strength with different anisotropic factors and ␣values. (a): ␣ = 3.1; (b): ␣ = 10/3; (c): ␣ = 11/3. Table 3 The maximum percentage difference of the irradiance scintillation index of optical waves between the anisotropic and isotropic turbulence cases derived in this work. ˛ = 3.1, ux = 1

˛ = 10/3, ux = 1

˛ = 11/3, ux = 1

uy = 2

uy = 3

uy = 4

uy = 2

uy = 3

uy = 4

uy = 2

uy = 3

uy = 4

2 I(pl) (˛, ux , uy )

26.60%

33.38%

36.18%

29.13%

36.01%

38.73%

33.32%

40.29%

42.88%

2 I(sp) (˛, ux , uy )

27.64%

34.61%

37.49%

29.82%

36.82%

39.57%

33.42%

40.40%

42.99%

Table 4 The maximum percentage difference of the irradiance scintillation index of optical waves between the anisotropic and isotropic turbulence cases derived in [22]. ˛ = 3.1

2 I(pl) (˛, ς ) 2 I(sp) (˛, ς )

˛ = 10/3

˛ = 11/3,

ς=2

ς=3

ς=4

ς=2

ς=3

ς=4

ς=2

ς=3

ς=4

56.40%

72.99%

80.69%

61.27%

77.69%

84.90%

68.52%

83.99%

90.09%

68.76%

116.01%

147.21%

62.24%

78.53%

85.56%

68.65%

84.08%

90.16%

position of irradiance scintillation index slightly moves to stronger turbulence. That is because the increased uy alleviates the multi-scattering effects of turbulence cells on the optical waves’ propagation. In addition, the maximum percentage differences of the irradiance scintillation index of optical waves derived in anisotropic and isotropic turbulence cases are calculated quantitatively and listed in Table 3. To make comparisons, the maximum percentage differences of the irradiance scintillation index of optical waves in anisotropic and isotropic turbulence derived in [22] are also calculated and listed in Table 4. Compared with the results shown in [22], the variable uy produces less effects on the irradiance scintillation index models of optical plane and spherical waves. Two anisotropic factors of ux and uy affect jointly the irradiance scintillation index of optical waves in anisotropic turbulence. 5. Discussions and conclusions In the modeling of the modified anisotropic turbulence refractive-index fluctuations spectra and the irradiance scintillation index of optical plane and spherical waves in moderate-to-strong anisotropic turbulence, the general distribution of asymmetric turbulence cells in the anisotropic turbulence is taken into considerations. Compared with the previously derived models in [22], the results derived in this work are no longer restricted to the assumption of circular symmetric distribution of turbulence cells in the plane orthogonal to the direction of propagation. Calculations indicate that both of the two anisotropic factors are indispensable in the analysis of optical waves’ propagation through anisotropic turbulence.

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