Investigation of temporal properties of optical wave irradiance fluctuations in anisotropic marine turbulence

Accepted Manuscript Title: Investigation of temporal properties of optical wave irradiance fluctuations in anisotropic marine turbulence Author: Linyan Cui PII: DOI: Reference:

S0030-4026(19)30733-8 https://doi.org/10.1016/j.ijleo.2019.05.081 IJLEO 62875

To appear in: Received date: Accepted date:

17 March 2019 23 May 2019

Please cite this article as: Cui L, Investigation of temporal properties of optical wave irradiance fluctuations in anisotropic marine turbulence, Optik (2019), https://doi.org/10.1016/j.ijleo.2019.05.081 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Investigation of temporal properties of optical wave irradiance fluctuations in anisotropic marine turbulence Linyan Cui1* School of Astronautics, Beihang University, Beijing 100191, China *[email protected]

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Abstract: Previous work on temporal statistical properties of optical waves in marine turbulence has focused on isotropic marine turbulence. Increasing studies indicate that anisotropy exists. This paper is the first work to investigate the temporal properties of plane and spherical waves in anisotropic marine turbulence. New temporal power spectral models for optical wave irradiance fluctuations will be developed. Comparisons between the new models and the published models for the

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anisotropic terrestrial turbulence will be conducted. Results exhibit the atmosphere humidity

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fluctuations aggravate the temporal variations of optical wave irradiance scintillation in contrast to the anisotropic terrestrial turbulence. In addition, the anisotropy of marine turbulence weakens the

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temporal fluctuations of optical wave irradiance.

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Keywords: marine turbulence, anisotropic turbulence, temporal power spectrum.

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1. Introduction

Temporal power spectra of optical wave reflect the optical wave temporal variations in the atmospheric

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turbulence media. Theoretical modeling for the optical wave temporal power spectra plays a key role in the performance evaluation and improvement of laser communication or imaging systems. The classical

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methods to establish the optical wave temporal power spectra are mainly focused on terrestrial turbulence or isotropic Kolmogorov marine turbulence [1-6]. For terrestrial turbulence, several refractive-index

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fluctuations spectra [6-11], have been established to investigate the optical wave temporal power spectra. General spectral power law, anisotropy of turbulence (only for anisotropic terrestrial turbulence), and

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finite inner and outer scales of turbulence, were taken into account. For marine turbulence, it supposes turbulence satisfy isotropic and Kolmogorov properties. The atmosphere temperature and humidity fluctuations, the finite outer and inner scales of marine turbulence were considered. Investigations indicate that the atmosphere humidity fluctuations strengthen marine turbulence’s impacts on the optical wave propagation compared with the terrestrial turbulence [12-14]. However, experiments and theoretical investigations indicate the significances of anisotropy and nonKolmogorov properties in marine turbulence [15-17]. First, the refractive-index fluctuations spectrum for

the anisotropic marine turbulence was proposed [16]. The anisotropy of marine turbulence, finite outer and inner scales of marine turbulence, and atmosphere refractive-index fluctuations induced by atmosphere temperature and humidity fluctuations are included in the proposed spectral model. Then, it was applied to investigate the OAM mode carried by partially coherent modified Bessel-Gaussian beams, and the scintillation and aperture averaging for Gaussian beam [16,17]. Results indicate that anisotropy in

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turbulence and general spectral power law are vital to analyze the above derived models. These researches aim at the optical wave spatial propagation properties. Until now, there is no investigation be performed about the optical wave temporal propagation properties in anisotropic marine turbulence.

In this work, new optical wave temporal power spectra for irradiance fluctuations will be developed. The atmosphere temperature and humidity fluctuations, the anisotropy in marine turbulence, the finite marine turbulence outer and inner scales, and the general spectral power law, will be taken into account during the theoretically modeling. Comparisons between the new derived models and those obtained for

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the anisotropic terrestrial turbulence will be conducted..

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2. Refractive-index fluctuations spectrum for anisotropic marine turbulence

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Marine environment exhibits different behaviors from the terrestrial turbulence [2,18], which brings a challenge to analyze the properties of optical wave propagation in such media. For anisotropic marine

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turbulence, the refractive-index fluctuations spectrum owns the expression as [16]: 3 /2    2     2 2  /2      0   n, Ma  ,  ,  , l0 , L0   A   C  1  a1  a2  exp   2    H    H   H 2

D

2 n

  . 

(1)

wavenumber

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where  H  c   / l0 is the wavenumber contributed by turbulence inner scale l0 ,  0  2 / L0 is the contributed

by

turbulence

outer

scale

L0 .

a1  0.061

and

a2  2.836 .

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   z2   2  x2   y2  .  is the parameter to characterize the degree of anisotropy in marine turbulence.

CC

The x, y, and z components for wavenumber   are described by  x ,  y , and  z . A   and c   are denoted by 1    (2) A    2    1 cos   , 4  2  1

A

   3   3      4         5   3  c     Aˆ        a   2   a   3  2  (3)  1    2     . 2  3  4  2        2 2  3   In the following modeling of optical wave temporal power spectra for irradiance fluctuations, the von  2  2 2  /2 Karman expression    0  exp   2  in Eq. (1) will be replaced by the exponential form of     H

  just like Andrew did in his work [2]. n,Ma  ,  ,  , l0 , L0  can be 

 n , Ma  ,  ,  , l0 , L0   A   C  2 n

2 





3 /2       1  a1   a2   H    H 

  2    2 1  exp  exp     2 2    0      H

  . 

(4)

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  2    2  1  exp   2   exp   2    0    H rewritten as 

3. Modeling of temporal power spectra for optical wave irradiance fluctuations in anisotropic marine turbulence

According to the classic work of Tatarskii [19], the optical wave temporal power spectra relate to the temporal covariance function by the Fourier transform. For the optical wave irradiance fluctuations, the

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temporal power spectra (it is also called power spectral density) WI   is defined as [19] 

0

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WI    4 CI  t  cos t  dt.

(5)

(6)

L    2 z  L  z   CI  sp   t   8 2 k 2    n   J 0  t  1  cos    d dz. Lk  0 0    

(7)

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M

A

L    2  L  z   CI  pl   t   8 2 k 2    n   J 0  t  1  cos    d dz, k  0 0   

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The optical wave temporal covariance function for irradiance fluctuations CI  t  is acquired from the optical wave spatial covariance function for irradiance fluctuations CI    with Taylor freezing

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assumption by variable substitution of   vt . In which,  is separation distance between points in the

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plane transverse to the direction of optical wave propagation.   is wind speed perpendicular to optical wave propagation path. k  2 /  and  is optical wavelength. J 0    is the Bessel function of zero

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order.

3.1 Plane wave temporal power spectrum for irradiance fluctuations in anisotropic marine turbulence

For anisotropic marine turbulence, the anisotropy of marine turbulence, finite outer and inner scales of marine turbulence, and general spectral power law should be discussed. Therefore, WI   will be restated with WI ,  ,  , l0 ,L0  . For plane wave, the following expression will be obtained. L

WI  pl   ,  ,  , l0 ,L0   32 2 k 2     n , Ma  ,  ,  , l0 , L0  0 0 0

(8)

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   2  L  z   J 0  t  1  cos    cos t  d dzdt. k    

By invoking Markov approximation, z component of   in n,Ma  ,  ,  , l0 , L0  can be neglected.   in

n,Ma  ,  ,  , l0 , L0  can be replaced with    . Rewriting cos . part in Eq. (8) with cos  x   Re  eix  and adopting the integration definition of [20]: 2 2 1/2   a  b  0 J 0  ax  cos  bx  dx   0  

0ba

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

N

ba

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WI  pl  ,  ,  , l0 ,L0  becomes

L 

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WI  pl   ,  ,  , l0 ,L0   32 2 k 2  0

  

(9)

 ,  ,  , l0 , L0 

/

2

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  1  cos   

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n , Ma

,

 L  z     2 2   2    

k





(10) 1/2

d dz ,

Making the integral for dz component in Eq. (10), Eq. (10) can be expressed as

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WI  pl   ,  ,  , l0 ,L0   32 2 k 2 L



  

n , Ma

 ,  ,  , l0 , L0 

/

 sin  L / k   1/2 1    2 2   2  d . 2  L/k   

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2

(11)

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 2 2 By making the variable substitution of t  2  1 , and adopting the integration definition of confluent 

hypergeometric function of the second kind U  a; c; z  [20]: U  a; c; z  



1 c  a 1 e zt t a 1 1  t  dt , a  0, Re( z )  0,   z 0

The analytic model for WI  pl  ,  ,  , l0 ,L0  is finally acquired.

(12)

32   1/ 2  A   Cˆ n2 2  2 k 2 L WI  pl   ,  ,  , l0 ,L0    v 1

     v 

(13)

 g pl (1)  ,  ,  , l0 , L0   g pl (2)  ,  ,  , l0 , L0   ,

g pl (i )  ,  ,  , l0 , L0  1

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2 k  1 3      1 1   exp   Ai  U  ; ; Ai    2  Im  exp   Bi U  ; ; Bi   2 2   v  2 2  L

1 2 a1     k  1 4      1 2      ; Ai    2  Im  exp   Bi U  ; ; Bi      exp   Ai  U  ;  H  v   2 2   v  2 2    L

a    2    H  v 

3-

 2

(14)

1 2  k  1 3        1 1      ; Ai    2  Im  exp   Bi U  ; ; Bi    . exp   Ai  U  ; 2 2 2   v  2     L

U

2   2  2   2 L , B   i ,   1 2 2 2  2 v   H  v   H k 2   2  2  2   2  2  A2  2  2  2  , B2  2  2  2  i  . v   H  0  v   H  0 

(15)

A

N

A1 

1/2

just like [3], in which, Fresnel frequency 0 is

and average wind speed  , the more convenient form of temporal

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formulated by Fresnel size  L / k 

1/2

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Making variable substitution of 0     L / k 

power spectrum for numerical analysis is obtained.

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32 1/ 2   R2 pl   ,     2  WI  pl   ,  ,  , l0 ,L0     2 1   0  0   g pl (1)  ,  ,  , l0 , L0   g pl (2 )  ,  ,  , l0 , L0   , 1/2  /2

EP

(16)

A

CC

in which,  R2 pl   ,   , g pl (1) ,  ,  , l0 , L0  and g pl (2) ,  ,  , l0 , L0  take the expressions as

 R2 pl   ,    1    A   Cˆ n2 2  2 k 3 /2 L /2        sin  ,  2  4 

1    4    

(17)

g pl i   ,  ,  , l0 ,L0  1

2   1 3      1 1   exp   Ai  U  ; ; Ai    2  Im exp   Bi  U  ; ; Bi   2 2   0  2 2   1 2    1 4      1 2      a1 A exp   Ai U  ; ; Ai    2  Im exp   Bi U  ; ; Bi    2 2   0  2 2      1/2 i

 a2 A

1 2  3  3 1    1    exp  A U ;3  ; A  Im exp  B U ; 2  ; B  i   i   i  i  .  2  2 4  2 4        0   

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3   2 4 i

(18)

3.2 Spherical wave temporal power spectrum for irradiance fluctuations in anisotropic marine turbulence

For anisotropic marine turbulence, WI   for spherical wave can be restated with WI ( sp ) ,  ,  , l0 ,L0  . L

U

WI  sp   ,  ,  , l0 ,L0   32 2 k 2     n , Ma  ,  ,  , l0 , L0 

N

0 0 0

(19)

A

   2 z  L  z   J 0  t  1  cos    cos t  d dzdt. Lk    

M

Using Eq. (9), the integration for dt is obtained. WI ( sp ) ,  , l0 ,L0  becomes

WI  sp   ,  ,  , l0 ,L0   32 k

EP

TE

D

2

Making variable substitution of t 

L 

2

   0

n , Ma

 ,  ,  , l0 , L0 

/

   2 z  L  z   2 2 2 1/2 1  cos     v    d dz. Lk    

(20)

 2v2  1 , and adopting the integral definition of U  a; c; z  , 2

A

CC

WI  sp ,  ,  , l0 ,L0  becomes

1 32 1/2  A   Cˆ n2 2  2 k 2 L    WI  sp   ,  ,  , l0 ,L0     v  v  1

  g , ,  , l , L   g sp (1)

0

0

sp (2)

 ,  ,  , l0 , L0  d ,

0

In which, g sp (1) ,  ,  , l0 , L0  and g sp (2) ,  ,  , l0 , L0  own the forms as

(21)

g sp (i)  ,  ,  , l0 , L0    1 3    1 3   exp   Ai  U  ; ; Ai   Re exp   Bi U  ; ; Bi   2 2  2 2   a1       1 4    1 4    ; Ai   Re exp   Bi U  ; ; Bi      exp   Ai  U  ;  H  v   2 2  2 2   

a    2   3v 2   

H

3-

 2

(22)

 3  3 1  1   ; Ai   Re exp   Bi  U  ;3  ; Bi    . exp   Ai  U  ;3  4 4 2  2    

A1 

2   2  v2   H2

SC RI PT



L 1      2   2 , B  i ,  1 2  2 v   H k  

L 1     2   2  2  2   2  2 A2  2  2  2  , B2  2  2  2  i . v   H  0  v   H  0 k  1/2

just like [3], and utilizing the detailed steps in

U

Making variable substitution of 0     L / k 

(23)

N

Appendix A, new analytic spherical wave temporal power spectral model in anisotropic marine turbulence is finally acquired.

32 1/ 2   R2 sp   ,     2  WI  sp   ,  ,  , l0 ,L0     2  2   0  0   g sp (1)  ,  ,  , l0 , L0   g sp (2 )  ,  ,  , l0 , L0   ,

M

A

1/2  /2

(24)

D

 R2 sp   ,     2    A   Cˆ n2 2  2 k 3 /2 L /2 ,        / 2   2    4   1    sin  .   2  4    

TE

2

A

CC

EP

1   g spi   ,  ,  , l0 ,L0   hsp (i)  d , c, z   a1  Ai1/2  hsp (i)  d , c  , z  2   3  2 4 i

 a2  A

3    hsp (i)  d , c  - , z  , 2 4  

  1  c   3 i 2  hsp (1)  a, c, z   exp   A1 U  a; c; A1   Re  F c  a ,1; c , ;   2 2 2 4 02     1  a  c  1   c  1   2  c    2   a    2  c  1/ 2 

1 c

 i 2   2   4 0 

 1 i  2   F 1  a ; 2  c  ;  , 1 1 2 4 02   

(25)

(26)

(27)

 1  c   2  hsp ( 2 )  a, c, z   exp   A2 U  a; c; A2   1 F1  c  a; c;  2 2   1  a  c  v k0     c  1   2       a   v2 k02 

1-c

3  . 2

(28)

(29)

SC RI PT

a  1/ 2, c 

 2  F 1  a ; 2  c ;  , 1 1 v2 k02  

4. Calculations and analysis

To remove dependence on Cˆ n2 , the scaled temporal power spectra 0WI ,  ,  , l0 ,L0  /  R2  ,   versus frequency ratio  / 0 will be calculated just like [2].   1.55 m , L  2000m . The impacts of anisotropy of marine turbulence will be discussed. Also, comparisons between the new derived models and those for the terrestrial turbulence will be conducted.

U

Figure 1 shows 0WI ,  ,  , l0 ,L0  /  R2   versus  / 0 under different anisotropic marine turbulence

N

conditions. l0 and L0 are set to 1mm and 50m. As shown, increased anisotropic degree of marine turbulence

A

enlarges the derived models. As shown, increased anisotropic degree of marine turbulence decreases the derived

M

models. The physical explanation for this phenomenon will be given as follows. When an optical wave propagates through the turbulence media, the turbulence cells will work as optical lens and they make the

D

optical wave deviates from the propagation direction. When the anisotropic factor  increases, the turbulence cells will act as optical lens with larger radius. Eventually, they will make optical waves

TE

deviate less from the propagation direction and focus better. In this case, the temporal fluctuations of optical wave irradiance will exhibit less severely with the increased  values.

EP

To investigate which turbulence cell scale has the greatest contribution to the optical wave temporal power spectra for irradiance fluctuations, WI ,  ,  , l0 ,L0  / 2 R2   as the function of  / 0 is

CC

plotted just like [2] and shown in Fig.2. Results show that the predominant frequency is close to

0 (  / 0  1 ) in weak marine turbulence. It indicates that in a given location along the optical wave

A

propagation path, the optimal turbulence cell which causes optical wave amplitude fluctuations takes the size nearly on the order of Fresnel scale ( L / k ). This conclusion is in accordance with the classic Rytov theory.

3.5

3.5 =1 =2 =3

3

plane w ave

2.5

spherical w ave

2.5

2

 0WI(sp) (  ,,,l0,L0)/R(sp) ( )

2

1.5

1

0.5

2

1.5

1

0.5

0 -1 10

0

0 -1 10

1

10

0

U

10  / 0 (a)

SC RI PT

2  0WI(pl) (  ,,,l0,L0)/R(pl) ( )

3

=1 =2 =3

10  / 0 (b)

1

10

N

Fig. 1. 0WI ,  ,  , l0 ,L0  /  R2   versus  / 0 in different anisotropic turbulence conditions

M

A

(  =3.8 ,  =1, 2,3 ). (a): For plane wave. (b): For spherical wave.

0.7

2  WI(sp) ( ,,,l0,L0)/2R(sp) ()

TE EP

0.4

0.2

A

0.5

peak

spherical w ave

0.4

0.3

0.2

0.1

0.1

0 -1 10

=1 =2 =3

0.6

plane w ave

0.5

0.3

D

peak

CC

2  WI(pl) ( ,,,l0,L0)/2R(pl) ( )

0.6

0.7

=1 =2 =3

0

10  / 0 (a)

1

10

0 -1 10

0

10  / 0 (b)

1

10

Fig. 2. WI ,  ,  , l0 ,L0  / 2 R2   versus  / 0 in different anisotropic turbulence conditions (  =3.8 ,  =1, 2,3 ). (a): For plane wave. (b): For spherical wave.

In Fig.3, both the derived temporal spectral models developed in marine and terrestrial turbulence conditions are plotted in different anisotropy degree of turbulence conditions. This figure will make the comparative analysis much clearer. 1

0.5

1

=1 plane w ave Marine

0.5

0.6

0.4

=2 plane w ave

0.2

Marine

Terres 0

10

0

10  / 0

2  0WI(sp) (  ,,,l0,L0)/R(sp) ( )

2

1

=1 spherical w ave

0.5

Marine Terres

TE 0

1

0.4

0.2

10

0 -1 10

Marine

0 -1 10

1

0

10  / 0

0.5

=2 spherical w ave Marine

0.4

0.3

0.2

=3 spherical w ave

0.1

Terres 0

10  / 0 (b)

1

10

Marine Terres

1

10

0 -1 10

0

10 / 0 (c)

1

10

EP

10  / 0 (a)

0.6

D

1.5

0.8

=3 plane w ave

0.1

10

N

1

2.5

0 -1 10

0 -1 10

A

3 2  0WI(sp) (  ,,,l0,L0)/R(sp) ( )

1

10  / 0

0.2

Terres

Terres

M

0 -1 10

0.3

U

1.5

0.4

SC RI PT

2

0.8

2  0WI(sp) (  ,,,l0,L0)/R(sp) ( )

2.5

2  0WI(pl) ( ,,,l0,L0)/R(pl) ( )

3 2  0WI(pl) ( ,,,l0,L0)/R(pl) ( )

2  0WI(pl) ( ,,,l0,L0)/R(pl) ()

3.5

Fig. 3. Comparison between the new derived models and those developed in terrestrial turbulence

CC

(  =3.8 ,  =1, 2,3 ). (a):   1 ; (b):   2 ;(c):   3 .

As shown, slightly more irradiance fluctuations are induced in marine turbulence media than those in

A

terrestrial turbulence especially when the anisotropic factor increases. This can be explained by the atmosphere refractive-index fluctuations, which are the essential attributes of optical wave irradiance fluctuations in turbulence media. For the marine turbulence, besides the temperature fluctuations in atmosphere, the humidity fluctuations also produce the atmosphere refractive-index fluctuations. The humidity fluctuations and the correlation between temperature and humidity fluctuations strengthen the refractive-index fluctuations. Eventually, the marine turbulence will produce more impacts on the optical

10

irradiance fluctuations. Additionally, when the anisotropy degree of turbulence exhibits more and more obvious, the marine turbulence will affect the derived models more severely than terrestrial discrepancy of the temporal power spectra between marine and terrestrial turbulence becomes larger and larger. In specific, the maximum differences of derived models in marine and terrestrial turbulence conditions are 5.97% (   1 ), 12.09% (   2 ), and 16.11% (   3 ) for plane wave. For spherical wave, they are 6.34%

SC RI PT

(   1 ), 12.50% (   2 ), and 16.38% (   3 ).

5. Conclusions and discussions

New analytic models for the optical wave temporal power spectra have been developed in the anisotropic marine turbulence. The anisotropy of turbulence, the humidity and temperature fluctuations, and the finite turbulence outer and inner scales are investigated in the new models. Compared with the known models

U

for the marine turbulence, the anisotropic factor of marine turbulence plays an important role for the investigations of optical wave temporal statistical property. Specifically, the increased anisotropy of

N

marine turbulence induces less effect on the new derived models. In addition, the anisotropic marine

A

turbulence induces more impacts to the optical wave temporal power spectra in comparison with the

M

previous ones developed for the terrestrial turbulence. That is because the humidity fluctuations in marine turbulence causes more atmosphere refractive-index variations, which eventually aggravate the temporal variations of optical wave at the receiver after propagating a long distance in the marine turbulence media.

D

The studies for the anisotropic marine turbulence are very important to investigate the optical imaging or

Acknowledgments

TE

laser communication in marine environment.

EP

This work is partly supported by the National Natural Science Foundation of China (61875003) and the

CC

Fundamental Research Fund for the Central Universities (YWF-19-BJ-J-125).

Appendix A

A

In this appendix, the integral for d in Eq. (21) will be solved. The key integration in Eq. (21) owns the 1

expression as  Re exp   Bi U  a; c; Bi   d , which can be solved with the following steps. 0

Firstly, utilizing the relationships of [20] U  a; c; z  

 1  c    c  1 1c z 1 F1 1  a  c; 2  c; z  1 F1  a; c; z    1  a  c   a

11

(A1)

e z 1 F1  a; c; z   1 F1  c  a; c;  z 

(A2)

exp   Bi U  a; c; Bi  can be rewritten as exp   Bi U  a; c; Bi  

 1  c 

 1  a  c 

F  c  a; c;  Bi 

1 1

(A3)

SC RI PT

  c  1 1c  Bi  1 F1 1  a; 2  c;  Bi   a

Secondly, expanding 1 F1  a; c; z  in the form of summarization of series [20]

 a n z n 1 F1  a; c; z    n 0  c n n ! 

1

i

i

becomes

U

 exp   B U  a; c; B  d 0

1

N

 exp   B U  a; c; B  d i

(A4)

i

0

n  Re  Bi  d 

D

 1  c    c  a n  1   c  n!  1  a  c  n 0 n

0

(A5)   c  1  1  a n  1    a  n 0  2  c n n !

n 1

 Re  B

1 c  n i

 d 

0

TE



n 1

M

A

1   1  c     c  1 1c   Re  Bi 1 F1 1  a; 2  c;  Bi   d  1 F1  c  a; c;  Bi    a 0   1  a  c  

EP

Substituting Bi into Eq. (A5) and making use of the definition of 2 F1  a, b; c; z  [20]

CC

1  c c b 1 a t b1  1  t   1  tz  dt , 2 F1  a, b; c; z     b    c  b 0

2

F1  a, b; c;1 

 c  c  a  b .  c  a   c  b

A

Finally, the integral for d in Eq. (A5) is acquired. Eq. (A5) can be written as

12

(A6)

(A7)

  1  c    c  a    2 L n 1 n n i 2  0 Re exp   B1 U  a; c; B1  d  Re   1  a  c   n  0  c  n n !  4v k   3 / 2  n  1

  c  1   2  c    1  a n  2L   i     a  2  2  c  1/ 2  n 0  2  c  1/ 2 n n!  4v2 k 

1 c  n

  

(A8)

  1  c    L  Re   2 F2  c  a,1; c,3 / 2; i 4v2 k    1  a  c     c  1   2  c     2 L    i    a  2  2  c  1/ 2   4v2 k 

SC RI PT

2

1 c

 exp   B2 U  a; c; B2  d  0

 1  c    2 2  F c  a ; c ;    1 1  1  a  c  v2 02     c  1   2 2      a   v2 02 

1-c

1

  2 2  F 1  a ; 2  c ;  . 1 1 v2 02  

(A9)

U

1

  2 L   F  1  a; 2  c  1/ 2; i 2   4v k   

1 1

N

Substituting  Re exp   Bi U  a; c; Bi   d into WI  sp  ,  , l0 ,L0  , the analytic expression for Eq. (21)

A

0

M

is obtained.

References

D

1. G.L.Siqueira and R.S.Cole, “Temporal-frequency spectra for plane and spherical waves in a millimetric

TE

wave absorption band,” IEEE Transactions on Antennas and Propagation, 39(2): 229-235, 1991. 2. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Optical Engineering Press, Bellingham, 2005).

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A

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SC RI PT

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U

13. Mingjian C F. S. Vetelino, K. Grayshan and C. Y. Young, “Inferring path average Cn2 values in the

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A

turbulence: long term beam spread and beam wander analysis,” Proc. SPIE, 7814, 7814R-1-10 (2010).

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15. M. Chang, C. O. Font, F. Santiago, Y. Luna, E. Roura, and S. Restaino, “Marine environment optical propagation measurements,” Proc. SPIE, 5550, 40-46 (2004).

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16. Mingjian Cheng, Lixin Guo, and Yixin Zhang, “Scintillation and aperture averaging for Gaussian

32606-32621 (2015).

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beams through non-Kolmogorov maritime atmospheric turbulence channels,” Optics Express, 23(25): 17. Yun Zhu, Ming Chen, Yixin Zhang, and Ye Li, “Propagation of the OAM mode carried by partially

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coherent modified Bessel–Gaussian beams in an anisotropic non-Kolmogorov marine atmosphere,” J. Opt. Soc. Am. A, 33(12): 2277- (2016).

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18. R. J. Hill, “Spectra of fluctuations in refractivity, temperature, humidity, and the temperature-humidity cospectrum in the inertial and dissipation ranges (atmospheric effects on radio propagation),” Radio

A

Science, 13, 953-961 (1978). 19. V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (trans. for NOAA by Israel Program for Scientific Translations, Jerusalem, 1971).

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