Theoretical expressions of long term beam spread and beam wander for Gaussian wave propagating through generalized atmospheric turbulence

Optik 126 (2015) 4704–4707

Contents lists available at ScienceDirect

Optik journal homepage: www.elsevier.de/ijleo

Theoretical expressions of long term beam spread and beam wander for Gaussian wave propagating through generalized atmospheric turbulence Linyan Cui ∗ , Lei Cao School of Astronautics, Beihang University, Beijing 100191, China

a r t i c l e

i n f o

Article history: Received 31 July 2014 Accepted 19 August 2015 Keywords: Atmospheric optics Non-Kolmogorov turbulence Beam spread Beam wander

a b s t r a c t Effects of the generalized atmospheric turbulence on the beam spread and beam wander for Gaussian wave are studied in detail both analytically and numerically. New analytic expressions of long term beam spread and beam wander are derived. They cover finite turbulence inner and outer scales and have a general spectral power law value in the range 3–4 instead of standard power law value of 11/3 (for Kolmogorov turbulence), and reduce correctly to the previously published analytic expressions for the case of Gaussian beam propagation through weak conventional Kolmogorov turbulence. The final results indicate that, turbulence inner and outer scales, and the general spectral power law value have significant influences on the Gaussian beam’s propagation. © 2015 Elsevier GmbH. All rights reserved.

1. Introduction Random temperature fluctuations of the atmosphere yield random refractive index fluctuations, which make the Gaussian beam deviates randomly from the direction of propagation. Serious turbulence effects are produced, including irradiance scintillation, angle of arrival fluctuations, beam wander, and beam spread. In this work, the latter two turbulence effects are analyzed. Traditionally, they are derived with the assumption that the turbulence is Kolmogorov type. Several turbulence spectral models have been developed, including Kolmogorov, Tatarskii, Von Karman, Exponential, and the modified atmospheric spectral models. These spectral models have been widely applied in investigating turbulence effects for Gaussian beam propagating through Kolmogorov turbulence. However, the turbulence effects models derived for the Kolmogorov turbulence cannot be directly applied in the generalized atmospheric turbulence (it is also called non-Kolmogorv turbulence), which has been demonstrated by experimental results [1–4] and theoretical investigations [5,6]. Non-Kolmogorov spectral model of refractive-index fluctuations has been proposed to investigate the beam wander and beam spread [7,8]. However, this spectral model does not consider the influences of turbulence

∗ Corresponding author. Tel.: +86 13401014803. E-mail address: [email protected] (L. Cui). http://dx.doi.org/10.1016/j.ijleo.2015.08.078 0030-4026/© 2015 Elsevier GmbH. All rights reserved.

inner and outer scales. To investigate the non-Kolmogorov atmospheric turbulence, some theoretical spectral models are also developed, such as the generalized von Karman spectral model [9] and the generalized exponential spectral model [10], which consider finite turbulence inner and outer scales and have general spectral power law values instead of the standard power law value of 11/3. Compared with other spectral models, the generalized modified atmospheric turbulence spectral model [11] proposed in our previous work can characterize the high frequency enhancement property of non-Kolmogorov turbulence, and has been used to analyze the irradiance scintillation for optical waves propagating through weak non-Kolmogorov turbulence [12–14]. In this study, the generalized modified atmospheric spectral model is used to investigate the long term beam spread and the beam wander for Gaussian beam propagating through weak nonKolmogorov turbulence. Numerical calculations are conducted to analyze the influences of turbulence inner scale, turbulence outer scale and the general spectral power law values on the derived expressions.

2. Generalized modified atmospheric spectral model for non-Kolmogorov turbulence The generalized modified atmospheric spectrum [11] is a non-Kolmogorov turbulence spectral model which considers the influence of finite turbulence inner and outer scales and has a general spectral power law value in the range of 3–5 instead of standard

L. Cui, L. Cao / Optik 126 (2015) 4704–4707

limited spot size radius, T(˛) is the long term beam spread which includes small-scale beam spreading and beam wander atmospheric effects, and given by [15]

power law value 11/3 for Kolmogorov turbulence. Specifically, it is given by [11] ˚n (, ˛, l0 , L0 ) ˆ · Cˆ n2 · −˛ · f (k, l0 , L0 , ˛), = A(˛)

 f (, l0 , L0 , ˛) =



1 − exp

 × exp

−

− 2

2

(0 ≤  < ∞,

  1 + a1 ·

02

 l



l2

− b1 ·

2 2

T (˛) = 4 k L · 0

ˆ A(˛) 

c(˛) =

+a1 · 



−

4 − ˛

˛ +2 2

3

−

 − b1 · 

˛ 3 + 2 2



−˛ + 3 + ˇ 2





 exp

3+ˇ−˛ 3



1 ˛−5

.

3 < ˛ < 5)

0

−

2 02

2 2

≈1+

 ˇ 

   l

1 + a1 ·

0  

− Lk



·

 1 − exp



 l

− b1 ·

L2  2 − k

(8)

 dd.

in Eq. (8) with Taylor series

∞ n  (−L2  2 /k)

n!

n=1

1 + a1 ·

−42 k2 L

 l

− b1 ·

ˆ · A(˛) · Cˆ n2 ·

(9)

,

  ˇ  l

∞ n  (−L/k) n=1 

−

exp

(5)

n!

2



l2

1 2n + 1

∞



1 − exp

1−˛+2n 0

−

2 02

 d. (10)

Using the definition of gamma function  (x) and the generalized hypergeometric function 2 F1 (a, b; c; z) [16]:

2

= W [1 + T (˛)].

∞

2 F1 (a, b; c; z)



−

2

+

3 2

(11)

∞  (a)n (b)n · z n

(c)n · n!

|z| < 1

,

(12)

The analytical expression of long term beam spread for Gaussian beam propagating through weak non-Kolmogorov turbulence with horizontal path is obtained, and it contains finite inner and outer scales, and has general spectral power law values. T (˛, l0 , L0 ) = T1 (˛, l0 , L0 ) − T2 (˛, l0 , L0 ),

  

 

1−



· 1 − 2 F1

 

−˛ + ˇ +1 2

=

n=0

ˆ T1 (˛, l0 , L0 ) = 22 k2 L · A(˛) · Cˆ n2 · kl2−˛ ·

 ˛

( > 0, x > 0)

0

where We is the spot size radius for Gaussian beam propagating through weak non-Kolmogorov turbulence, W is the diffraction

+a1 · 

x−1 · e− d,

 (x) =

(6)

b1 · 



L2  2 − k



The beam spot size of a Gaussian beam propagating in turbulence is affected from two main effects: beam spread and beam wander. The combined effect of beam wander and beam spread is called the long term beam spread (represents the effective beam spot size), and it is used as one of the main parameters to evaluate the intensity profile along the path. The beam spot size for a Gaussian beam propagating through weak non-Kolmogorov turbulence can be expressed as [15] =

 1 − exp

l2

T (˛, l0 , L0 ) =

3. Long term beam spread for Gaussian beam in non-Kolmogorov turbulence

2 WLT (˛)





1−˛

and then making the integration with respect to the parameter of , Eq. (8) becomes

when ˛ = 11/3, the generalized modified atmospheric spectral model is reduced to the Kolmogorov modified spectral model, and when l0 → 0, L0 → ∞ , it becomes the general non-Kolmogorov spectral model ˆ ˚n (, ˛) = A(˛) · Cˆ n2 · −˛ . (0 ≤  < ∞,

2

Expanding exp

(4)

We2

−

(3)

3−˛ 3



1 ∞ ˆ T (˛, l0 , L0 ) = 4 k L · A(˛) · Cˆ n2

exp



2L . kW2

2 2



 

dd.

To consider the influences of finite turbulence inner scale, finite turbulence outer scale, and general spectral power law values, T(˛) will be replaced by T(˛, l0 , L0 ), and the theoretical expression of long term beam spread for Gaussian beam propagating through weak non-Kolmogorov turbulence with horizontal path will be derived. When the spectral power law ˛ equals 4, there exists singularity in the calculations, in this study, ˛ is restricted to the range 3–4 for analysis convenient just as [7,8]. Substituting Eq. (1) into Eq. (7), T(˛) becomes

(2)

is the generalized refractive-index structure parameter with unit m3−˛ . The coefficients of a1 , b1 and ˇ in the generalized modified ˆ atmospheric spectrum depend on experimental results. A(˛) and c(˛) take the forms as [11]





0

where  = 1 − z/L,  =

ˆ (˛) =  (˛ − 1) sin (˛ − 3)  , A 2 42

L2  2 − k

(7)

l

.



 · ˚n (, ˛) 1 − exp

  ˇ 

ˆ where ˛ is the general spectral power law, A(˛) is a constant which maintains consistency between the refractive index structure function and its power spectrum,  is the angular spatial frequency with units of rad/m, l = c(˛)/l0 , 0 = C0 /L0 . l0 and L0 are turbulence inner and outer scales, respectively. The choice of C0 depends on the specific application, and it is set to 4 in this study just as [15]. Cˆ n2





1 ∞

(1)

3 < ˛ < 5)

4705

· 1 − 2 F1



  ˛ 2

· 1 − 2 F1



1− 2

−

Ll ˛ 3 1 3 + , ; ;− 2 2 2 2 k

 2

(13)

Ll ˛ 1 3 , ; ;− 2 2 2 k



− 2

Ll 1 3 −˛ + ˇ + 1, ; ; − 2 2 2 k

(14)

 ,

4706

L. Cui, L. Cao / Optik 126 (2015) 4704–4707

To consider the influences of finite turbulence inner scale, finite turbulence outer scale, and general spectral power law values, rc2  will be replaced by rc2 (˛, l0 , L0 ), and theoretical expression of the variance of beam wander for Gaussian beam propagating through weak non-Kolmogorov turbulence with horizontal path will be derived. When the spectral power law ˛ equals 4, there exists singularity in the calculations, in this study, ˛ is restricted to the range 3–4 for analysis convenient just as [7,8]. Substituting Eq. (1) into Eq. (16), rc2  becomes

L ∞ rc2 (˛, l0 , L0 ) = 42 k2 W 2 Fig. 1. Numerical calculations for the long term beam spread. (a) The long term beam spread as a function of ˛ with different l0 values. (b) The long term beam spread as a function of l0 with different ˛ values.

 ˛ k22

ˆ T2 (˛, l0 , L0 ) = 22 k2 L · A(˛) · Cˆ n2 · ˛ 3 − 2 · −1 ·  2 +a1 · k2 l ˛−ˇ −1 −ˇ · l ·  b1 · k2 2



 ˛ −

2

+

−1

·



1−

˛ 2

 

3 2

 

 ˛

· 1 − 2 F1 −

 

−˛ + ˇ +1 2



· 1 − 2 F1



· 1 − 2 F1 1 −

2

+

L ∞ = 4 k W

˚n ()HLS (, z)(1 − e−L

2 0

2  2 /k

)ddz.

where  = 1 − z/L, HLS (, z) is the large-scale filter function [17–19]

= exp



2

2

(18)



−

(15)

⎫ ⎬ .

⎭

The gauss filter function in Eq. (18) only permits turbulence cells with the size equal to the beam size and larger to contribute to the beam wander, and small-scale turbulence’s effect can be eliminated. The turbulence outer scale forms an upper bound on the inhomogeneity size that can cause beam wander, and turbulence inner scale’s effect on the beam wander is negligible [15]. Therefore, only the turbulence outer scale parameter in the generalized modified atmospheric spectral model is considered, and Eq. (18) becomes



L ∞ ˆ rc2 (˛, l0 , L0 ) = 42 k2 W 2 A(˛) · Cˆ n2



1−˛ 1 − exp 0

· HLS (, z)(1 − e−L

−

0

2  2 /k

2



02

)ddz.

(19)

HLS (, z) = exp



−2 W02



0 + 0 

2



+ 20 1 − 

2

.

(20)

To emphasize the refractive nature of beam wander, the last term in Eq. (20) is dropped [15]. With the geometrical optics approximation of 1 − e−L

2  2 /k

≈

L2  2 , k

L2

1 k

(21)

and adopting the definition of 2 F1 (a, b; c; z) function [16]:

−2 W02 [(1 − z/F0 ) + (2z/kW02 ) ]



)ddz.

(16)

0

HLS (, z) = exp [−2 W 2 (z)]

˛ 1 3 L , ; ;− 2 2 2 k · 2

2  2 /k

For integration purpose, HLS (, z) is expressed with the beam parameters (0 and 0 ) [15]

Beam wander describes the displacement of the instantaneous center of the beam in the receiver plane, and the variance of beam wander takes the form as [15]

2 2

0

× (, z)(1 − e−L

1 3 L −˛ + ˇ + 1, ; ; − 2 2 2 k · 2

4. Beam wander for Gaussian beam in non-Kolmogorov turbulence

rc2 

0

L 3 1 3 , ; ;− 2 2 2 k · 2

Fig. 1 analyzes the influences of ˛ and l0 ’s variations on the long term beam spread. The parameters are set as: L = 1000 m, Cˆ n2 = 7 × 10−14 m3−˛ ,  = 1.55 ␮m, W0 = 1 cm. Fig. 1(a) shows the long term beam spread as the function of ˛. It can be seen that if ˛ decreases from ˛ = 11/3 (excluding alpha values close to 3), the long term beam spread increases up to a maximum value. At this point ˆ the curve changes its slope because of the term A(˛) that assumes very low values. In addition it is shown that if ˛ increases from ˛ = 11/3, the long term beam spread decreases. Fig. 1(b) shows the long term beam spread as the function of l0 . As shown, when l0 increases, the value of long term beam spread firstly increases and then decreases, and the bump property introduced by the turbulence inner scale is exhibited. This phenomenon can be explained directly from the definition of the generalized modified spectral model: the bump in the refractive index power spectrum, which appears when the product of the wave number with inner scale is around unity ( · l0 ∼ 1), produces a corresponding bump in the long term beam spread.

˚n (, ˛)HLS

.

(17) 2 F1 (a, b; c; z) =

W0 is the effective beam radius at the transmitter, F0 is the phase curvature parameter of the Gaussian beam at the transmitter.

 (c)  (b) ·  (c − b)

1

t b−1 · (1 − t)c−b−1 · (1 − tz)−a dt,

0

(22)

L. Cui, L. Cao / Optik 126 (2015) 4704–4707

4707

propagating through weak non-Kolmogorov turbulence. Calculations show that turbulence inner scale produces obvious impacts on the long term beam spread for the non-Kolmogorov turbulence. As the turbulence inner scale increases, the long term beam spread value firstly increases and then decreases, and the bump property introduced by the turbulence inner scale is well exhibited. Calculations also show that the turbulence outer scale produces obvious impacts on the variance of beam wander for the non-Kolmogorov turbulence. As the turbulence outer scale increases, the variance of beam wander increases. Different spectral power law values bring obvious effect on the final expressions. The atmospheric turbulence’s influence on the optical wave propagation varies at different atmosphere layer and follows the rules derived in this study. Fig. 2. Numerical calculations for the variance of beam wander. (a) The variance of beam wander as a function of ˛. (b) The variance of beam wander as a function of l0 .

This work is partly supported by the National Natural Science Foundation of China (61405004) and the China Scholarship Council (201506025046).

the variance of beam wander becomes rc2 (˛, l0 , L0 )

= 4  2

⎧ ⎨

1 2 F1 (4 − ˛, 1; 4; 1 − 0 ) − ⎩3



˛ ˆ 2− · A(˛) · Cˆ n2 L3 · W0˛−4 · 2



1

kL2 · W02

2 1+ 0

kL2

·

W02 (0

+ 0 )

⎫ 2−˛/2 ⎬ ⎭

2

(23)

4 2   3







· 1−

rc2 foc (˛, l0 , L0 ) = 42 

1 ˛−1

−

2−

˛ 2



ˆ (˛) · Cˆ n2 L3 · W ˛−4 ·A 0

2−˛/2 

kL2 · W02 1 + kL2 · W02



2−



˛ 2



.

(24)

ˆ · A(˛) · Cˆ n2 L3 · W0˛−4 ·

˛ 3 5 1 2 2−˛/2 (k · W02 ) , ; ; −kL2 · W02 2 F1 2 − 3 L 2 2 2

References

.

Eq. (23) is applicable for collimated, divergent, or convergent (focused) Gaussian beam waves. For collimated beam (0 = 1) and focused beam (0 = 0), rc2  can be expressed with the analytical expressions rc2 col (˛, l0 , L0 ) =

Acknowledgments



(25) .

Fig. 2 analyzes ˛ and L0 ’s variations on the variance of beam wander for both collimated beam and focused beam cases. The parameters are set as: L = 1000 m, Cˆ n2 = 7 × 10−14 m3−˛ ,  = 1.55 ␮m, W0 = 1 cm. It can be seen that in the case of collimated beam, the variance of beam wander for a focused beam can be greatly diminished in the presence of a finite outer scale. Fig. 2(a) shows the variance of beam wander as the function of ˛. As shown, when ˛ decreases from ˛ = 11/3(excluding alpha values close to 3), the variance of beam wander increases up to a maximum value. At ˆ that this point the curve changes its slope because of the term A(˛) assumes very low values. Fig. 2(b) shows the variance of beam wander as the function of L0 . As shown, the variance of beam wander increases with the increased L0 value. It can be explained from this point of view: the beam wander is mainly caused by the large-scale turbulence cell. When L0 increases, according to the Richardson cascade theory, the wave meets a major number of large-scale turbulent cells along its propagation length and these cells lead to higher variance of beam wander with respect to the case of lower outer scale value, where more large scales are cut out. 5. Conclusions In this study, new analytic expressions for the long term beam spread and the beam wander are derived for Gaussian beam

[1] D.T. Kyrazis, J.B. Wissler, D.B. Keating, A.J. Preble, K.P. Bishop, Measurement of optical turbulence in the upper troposphere and lower stratosphere, Proc. SPIE 2120 (1994) 43–55. [2] M.S. Belen’kii, S.J. Karis, J.M. Brown, R.Q. Fugate, Experimental study of the effect of non-Kolmogorov stratospheric turbulence on star image motion, Proc. SPIE 3126 (1997) 113–123. [3] M.S. Belen’kii, E. Cuellar, K.A. Hughes, V.A. Rye, Experimental study of spatial structure of turbulence at Maui Space Surveillance Site (MSSS), Proc. SPIE 6304 (2006), 63040U. [4] A. Zilberman, E. Golbraikh, N.S. Kopeika, A. Virtser, I. Kupershmidt, Y. Shtemler, Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence, Atmos. Res. 88 (2008) 66–77. [5] A.S. Gurvich, M.S. Belen’kii, Influence of stratospheric turbulence on infrared imaging, J. Opt. Soc. Am. A 12 (1995) 2517–2522. [6] M.S. Belen’kii, Effect of the stratosphere on star image motion, Opt. Lett. 20 (1995) 1359–1361. [7] I. Toselli, L.C. Andrews, R.L. Phillips, V. Ferrero, Free space optical system performance for laser beam propagation through non Kolmogorov turbulence, Proc. SPIE 6457 (2007), 64570T. [8] Y.X. Zhang, C.F. Si, Y.G. Wang, J.Y. Wang, J.J. Jia, Capacity for non-Kolmogorov turbulent optical links with beam wander and pointing errors, Opt. Laser Technol. 43 (2011) 1338–1342. [9] I. Toselli, L.C. Andrews, R.L. Phillips, V. Ferrero, Angle of arrival fluctuations for free space laser beam propagation through non Kolmogorov turbulence, Proc. SPIE 6551 (2007) 65510E. [10] L.-Y. Cui, B.-D. Xue, X.-G. Cao, J.-K. Dong, J.-N. Wang, Generalized atmospheric turbulence MTF for wave propagating through non-Kolmogorov turbulence, Opt. Express 18 (20) (2010) 21269–21283. [11] B. Xue, L. Cui, W. Xue, X. Bai, F. Zhou, Generalized modified atmospheric spectral model for optical wave propagating through non-Kolmogorov turbulence, J. Opt. Soc. Am. A 28 (5) (2011) 912–916. [12] C. Ji, L.1. Xu, Scintillation index and performance analysis of wireless optical links over non-Kolmogorov weak turbulence based on generalized atmospheric spectral model, Opt. Express 19 (20) (2011) 19067–19077. [13] L.Y. Cui, B.D. Xue, W.F. Xue, X.Z. Bai, X.G. Cao, F.G. Zhou, Expressions of the scintillation index for optical waves propagating through weak non-Kolmogorov turbulence based on the generalized atmospheric spectral model, Opt. Laser Technol. 44 (2012) 2453–2458. [14] X. Yi, Z.J. Liu, P. Yue, Optical scintillations and fade statistics for FSO communications through moderate-to-strong non-Kolmogorov turbulence, Opt. Laser Technol. 47 (2013) 199–207. [15] L.C. Andrews, R.L. Phillips, Laser Beam Propagation through Random Media, SPIE Optical Engineering Press, Bellingham, 2005. [16] L.C. Andrews, Special Functions of Mathematics for Engineers, 2nd ed., SPIE Optical Engineering Press, Bellingham, Wash., 1998. [17] V.I. Klyatskin, A.I. Kon, On the displacement of spatially bounded light beams in a turbulent medium in the Markovian-random-process approximation, Radiofiz. Quantum Electron. 15 (1972) 1056–1061. [18] V.L. Mironov, V.V. Nosov, On the theory of spatially limited light beam displacements in a randomly inhomogeneous medium, J. Opt. Soc. Am. 67 (1977) 1073–1080. [19] L.C. Andrews, et al., Beam wander effects on the scintillation index of a focused beam, Proc. SPIE 5793 (2005).