log-amplitude variance for a gaussian-beam wave propagating through non-kolmogorov turbulence

Log-amplitude variance for aGaussian-beam wave propagating

through non-Kolmogorov turbulence

Liying Tan1, Wenhe Du1 ,2 ∗ , Jing Ma1, Siyuan Y u1, and Qiqi Han1

1National Key Laboratory of Tunable Laser Technology, Harbin Institute of Technology,Harbin 150001, China

2College of Science, Qiqihar University, Qiqihar 161006, China

[email protected]

Abstract: In the past decades, both the increasing experimental evidencesand some results of theoretical investigation on non-Kolmogorov turbulencehave been reported. This has prompted the study of optical propagationin non-Kolmogorov atmospheric turbulence. In this paper, using a non-Kolmogorov power spectrum which owns a generalized power law insteadof standard Kolmogorov power law value 11/3 and a generalized amplitudefactor instead of constant value 0.033, the log-amplitude variances fora Gaussian-beam wave are derived in the weak-fluctuation regime for ahorizonal path. The analytic expressions are obtained and then used toanalyze the effect of spectral power-law variations on the log-amplitudefluctuations of Gaussian-beam wave.

© 2010 Optical Society of America

OCIS codes: (010.1290) Atmospheric optics; (010.1330) Non-Kolmogorov turbulence;(010.1330) Gaussian beam; (010.1300) Scintillation.

References and links1. M. S. Be1en’kii, S. J. Karis, J. M. Brown, and R. Q. Fugate, “Experimental evidence of the effects of non-

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wave-front generator,” Proc. SPIE6027, 602716-1-12 (2006).5. A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol

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6. A. Zilberman, E. Golbraikh, and N. S. Kopeika,“Lidar studies of aerosols and non-Kolmogorov turbulence in theMediterranean troposphere,” Proc. SPIE5987, 598702-1-12 (2005).

7. G. K. Batchelor, “Small-scale variation of convected quantities like temperature in turbulent fluid. Part I. Generaldiscussion and the case of small conductivity,” J. Fluid Mech. 5, 113-133 (1959).

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9. S. S. Moiseev and O. G. Chkhetiani, “Helical scaling in turbulence,” JETP83, 192-198 (1996).10. T. Elperin, N. Kleeorin, and I. Rogachevskii, “Isotropic and anisotropic spectra of passive scalar fluctuations in

turbulent fluid flow,” Phys. Rev. E53, 3431-3441 (1996).11. R. R. Beland, “Some aspects of propagation through weak isotropic non-Kolmogorov turbulence,” Proc. SPIE

2375, 1111-1126 (1995).

(C) 2010 OSA 18 January 2010 / Vol. 18, No. 2 / OPTICS EXPRESS 451#117361 - $15.00 USD Received 21 Sep 2009; revised 2 Nov 2009; accepted 8 Dec 2009; published 4 Jan 2010

12. B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmosphericturbulence,” Proc. SPIE2471, 181-196 (1995).

13. G. D. Boreman and C. Dainty, “Zernike expansions for non-Kolmogorov turbulence,” J. Opt. Soc. Am. A13,517-522 (1996).

14. C. Rao, W. Jiang, and N. Ling, “Spatial and temporal characterization of phase fluctuations in non-Kolmogorovatmospheric turbulence,” J. Mod. Opt.47, 175-177 (2000).

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16. M. S. Belen’kii, “Effect of the stratosphere on star image motion,” Opt. Lett.20, 1359-1361 (1995).17. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero,“Free space optical system performance for laser beam

propagation through Non-Kolmogorov turbulence,” Proc. SPIE 6457, 64570T-1-11 (2007).18. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero,“Angle of arrival fluctuations for free space laser beam

propagation through Non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E-1-12 (2007).19. W. B. Miller, J. C. Ricklin, and L. C. Andrews, “Log-amplitude variance and wave structure function: a new

perspective for Gaussian beams,” J. Opt. Soc. Am. A10, 661-672 (1993).20. L. C. Andrews and R. L. Phillips,Laser Beam Propagation through Random Media, (SPIE Optical Engineering

Press, Bellingham, 1998).21. M. Abramowitz and I. A. Stegun,Handbook of Mathematical Functions, (Dover Publications INC., New York,

1965).

1. Introduction

At present, it has been accepted that the Kolmogorov model isnot the only possible turbulentone in the atmosphere, which is supported by numerous experimental evidences [1-6] and someresults of theoretical investigation [7-10]. This has prompted the investigation of optical wavepropagation through the atmospheric turbulence exhibiting non-Kolmogorov statistics.

Beland analyzed a representative amplitude effect, the log-amplitude variance, and a phaseeffect, the coherent length, when the refractive-index fluctuations deviate from Kolmogorovstatistics [11]. Stribling et al defined the turbulence, in which the structure function for the in-dex of refraction and the corresponding power spectrum obeyed an arbitrary power law, as non-Kolmogorov turbulence and presented an analysis of opticalpropagation in non-Kolmogorovatmospheric turbulence, mainly including the wave structure function, the Strehl ratio, etc [12].Boreman and Dainty studied the expressions of non-Kolmogorov turbulence in the light ofZernike polynomials [13]. Rao et al analyzed the spatial andtemporal characterization of phasefluctuations in non-Kolmogorovatmospheric turbulence [14]. Gurvich and Belen’kii introduceda model for the power spectrum of stratospheric non-Kolmogorov turbulence and investigatedthe stratospheric turbulence on the scintillation and the coherence of starlight and on the degra-dation of star image [15]. Belen’kii studied the influence ofthe stratosphere on star image mo-tion again based on the model for the power spectrum of stratosphere [16]. Recently, Toselliaet al presented a non-Kolmogorov theoretical power spectrum model and estimated free spaceoptical system performance for laser beam propagating horizontally through non-Kolmogorovatmospheric turbulence [17]. And then they analyzed the angle-of-arrival fluctuations for freespace laser beam again [18]. So far all of theoretical investigations for optical propagation innon-Kolmogorov atmospheric turbulence have focused on theunbounded plane or spherical-wave models. However, in many applications the plane and spherical-wave approximations donot suffice to describe the propagation properties of optical wave. Therefore, it is very necessaryto extend the investigation of optical propagation in non-Kolmogorov atmospheric turbulenceto Gaussian-beam wave model.

In this paper, a non-Kolmogorov theoretical power spectrumis considered [17], which has ageneralized power law that takes all the values ranging from3 to 4. As the power lawα is setto the standard Kolmogorov value 11/3, the spectrum reducesto the conventional Kolmogorovone. Based on this spectrum and following the same procedurealready used from Miller etal in the Kolmogorov case [19], the log-amplitude variancesfor a Gaussian-beam wave have


been derived in weak turbulence for a horizonal path and the influence of spectral power-lawvariations on the log-amplitude fluctuations is analyzed.

2. Non-Kolmogorov spectrum

For the purpose of this paper, a theoretical power spectrum model that describes non-Kolmogorov optical turbulence is considered [17], which obeys a more general power law andin which the power-law exponents can take all the values ranging from 3 to 4,

Φn(κ ,α) = A(α) C̃2n κ−α , 2π/L0 ≪ κ ≪ 2π/l0, 3 < α < 4, (1)

whereκ is the magnitude of three dimensional wave number vector (inunits of rad/m),α is thespectral power-law exponent,̃C2

n is a generalized refractive-index structure parameter (inunitsof m3−α) that describes the strength of the turbulence along the path, l0 andL0 denote the innerand outer scales of turbulence, respectively, andA(α) is a function defined by

A(α) =1

4π2 Γ(α −1) cos(απ

2

)

, (2)

where the symbolΓ(x) represents the gamma function. When the power lawα is equal to 11/3,A(11/3)= 0.033,C̃2

n =C2n , and the spectrum reduces to the conventional Kolmogorov spectrum

[20],Φn(κ) = 0.033C2

n κ−11/3, (3)

whereC2n represents the conventional refractive-index structure parameter and has units of

m−2/3. In addition, asα → 3, A(α) → 0. As a result, the power spectrum for refractive-indexfluctuations disappears in the limiting caseα = 3. Finally, it can be seen from Eq. (1) thatall of the analyses performed in this paper are only related to the inertial interval of turbulentspectrum, i.e., 2π/L0 ≪ κ ≪ 2π/l0.

3. Log-amplitude variance

Although our study is concerned with non-Kolmogorov turbulence, we still start our work fromthe conventional Kolmogorov results. In the weak-fluctuation regime the log-amplitude vari-ance for a Gaussian-beam wave propagating through the conventional Kolmogorov turbulenceis given by [20]

σ2χ(ρ) = 2π2k2L

∫ 1

0

∫ ∞

0κΦn(κ)

{

I0(2Λρξ κ)−cos

[

Lκ2

k(1− Θ̃ξ )ξ

]}

×

[

exp

(

−ΛLξ 2κ2

k

)]

dκdξ , (4)

whereρ is distance from the beam center line in the plane transverseto the propagation di-rection (z axis), k = 2π/λ andλ is the optical wavelength,L is the propagation distance ofthe beam in the turbulent atmosphere,Φn(κ) is the power spectrum for refractive-index fluc-tuations,I0(x) is a modified Bessel function of the first kind,ξ is related toz by z = 1− ξ/L,and the complementary parameterΘ̃ = 1−Θ. Θ andΛ are the output plane (or receiver) beamparameters that are related to the input plane (or transmitter) beam parametersΘ0 andΛ0 by

Θ =Θ0

Θ20 + Λ2

0

, Λ =Λ0

Θ20 + Λ2

0

, (5)

whereΘ0 andΛ0 are defined byΘ0 = 1− L/R0 andΛ0 = λ L/πW20 , respectively. Here,W0

andR0 denote the radius of the beam size and the radius of curvatureof the phase front at the


transmitter. The parameterΘ0 is also called the curvature parameter andΛ0 is called the Fresnelratio at the input plane, while the quantityΘ is called the curvature parameter andΛ is calledthe Fresnel ratio at the output plane. Either the transmitter beam parameters or the receiverbeam parameters can describe the diffractive characteristics of Gaussian-beam wave.

For interpretation purposes, the log-amplitude variance is usually expressed as the sum ofthe radial component,σ2

χ ,r(ρ), and the longitudinal component,σ2χ ,l ,

σ2χ(ρ) = σ2

χ ,r(ρ)+ σ2χ ,l, (6)

where

σ2χ ,r(ρ) = 2π2k2L

∫ 1

0

∫ ∞

0κΦn(κ) [I0(2Λρξ κ)−1]

×

[

exp(−ΛLξ 2κ2

k)

]

dκdξ , (7)

σ2χ ,l = 4π2k2L

∫ 1

0

∫ ∞

0κΦn(κ)sin2

[

Lκ2

2k(1− Θ̃ξ )ξ

]

×

[

exp(−ΛLξ 2κ2

k)

]

dκdξ . (8)

The radial component physically denotes the off-axis contribution to the log-amplitude varianceand vanishes at the beam center line (ρ = 0) or asΛ = 0 (corresponding to an infinite wave suchas a plane or spherical wave), whereas the longitudinal component is constant throughout thebeam cross section in any transverse plane.

It is noteworthy that the explicit form of the refractive-index power spectrum is not takenin the development of the above equations. Therefore, when the Kolmogorov power spec-trum is substituted into Eq. (4), the conventional results will be obtained. Here, using a non-Kolmogorov power spectrum and following the same procedureas discussed in Ref. [17] forthe standard Kolmogorov spectrum, the log-amplitude fluctuations for a Gaussian-beam wavepropagating in non-Kolmogorov atmospheric turbulence areanalyzed.

For the radial component, our analysis results in

σ2χ ,r(ρ ,α) =

1α −1

A(α) C̃2nπ2k3−α/2Lα/2Γ

(

2−α2

)

Λα−2

2

×

[

1F1

(

2−α2

;1;2ρ2

W 2

)

−1

]

, (9)

where1F1(a;c;z) is the confluent hypergeometric function of the first kind andW is the beamradius at the receiver. When the relationρ ≤ W or ρ ≫ W is satisfied, based on the first fewterms of the small-argument series representation or the large-argument asymptotic form of the1F1 function, the approximate results are also obtained,

σ2χ ,r(ρ ,α) =

2α −1

A(α) C̃2nπ2k3−α/2Lα/2Λ

α−22

ρ2

W 2

×

[

Γ(

4−α2

)

+ Γ(

6−α2

)

ρ2

2W 2

]

, ρ ≤W, (10)

σ2χ ,r(ρ ,α) =

1α −1

2−α/2A(α) C̃2nπ2k3−α/2Lα/2Λ

α−22

×

(

Wρ

)αexp

(

2ρ2

W 2

)

, ρ ≫W. (11)


For the longitudinal component, the corresponding result is expressed as

σ2χ ,l(α) =

1α −1


(

2−α2

)

{

Λα−2

2

−Re

[

2α −2α

iα−2

2 2F1

(

2−α2

,α2

;2+ α

2;(Θ̃ + iΛ)

)]}

, (12)

where2F1(a,b;c;z) is the hypergeometric function and Re means the real part of the expressionin the square brackets.

Finally, the log-amplitude variance for a Gaussian-beam wave propagating through non-Kolmogorov turbulence is given by

σ2χ(ρ ,α) =

1α −1


(

2−α2

){

Λα−2

2 1F1

(

2−α2

;1;2ρ2

W 2

)

−Re

[

2α −2α

iα−2

2 2F1

(

2−α2

,α2

;2+ α

2;(Θ̃ + iΛ)

)]}

. (13)

Here it is noted that the assumptions that non-Kolmogorov turbulence is homogeneous alongthe propagation path and the inner scale is much smaller thanthe size of the Fresnel zone areinvolved in the above evaluations.

As for the Kolmogorov case, for the special beam form for the non-Kolmogorov case, thehypergeometric function2F1 in Eqs. (12) and (13) can also be simplified to more tractableanalytic functions. For example, for|Θ̃ + iΛ| ≤ 1 (asΘ0 ≥ 0.5), which corresponds to all di-vergent and collimated beam and some convergent beam models, the series denotation of thehypergeometric function leads to

Re

[

2α −2α

iα−2

2 2F1

(

2−α2

,α2

;2+ α

2;(Θ̃+ iΛ)

)]

=

2α −2α

∞

∑n=0

(2−α2 )n(

α2 )n

(2+α2 )nn!

(Θ̃2 + Λ2)n/2cos

[

n tan−1(

ΛΘ̃

)

+(α −2)π

4

]

, (14)

where(a)n = Γ(a + n)/Γ(a), n = 0,1,2, · · ·. Moreover, for|Θ̃ + iΛ| > 1 (asΘ0 < 0.5), usingthe analytic continuation formula of the hypergeometric function [21],

2F1(a,b;c;y) =Γ(c)Γ(b−a)

Γ(b)Γ(c−a)(−y)−a

2F1

(

a,1− c + a;1−b+a;1y

)

+Γ(c)Γ(a−b)

Γ(a)Γ(c−b)(−y)−b

2F1

(

b,1− c + b;1−a+b;1y

)

, (15)

yields

Re

[

2α −2α

iα−2

2 2F1

(

2−α2

,α2

;2+ α

2;(Θ̃+ iΛ)

)]

= (Θ̃2 + Λ2)α−2

4

∞

∑n=0

(2−α2 )n(1−α)n

(2−α)nn!(Θ̃2 + Λ2)−n/2cos

[(

n−α −2

2

)

× tan−1(

ΛΘ̃

)

+(α −2)π

4

]

+2α −2

αΓ(1+ α/2)Γ(1−α)

Γ(1−α/2)(Θ̃2 + Λ2)−

α4

×cos

[

α2

tan−1(

ΛΘ̃

)

+(2−3α)π

4

]

. (16)


Table 1. Expressions of Log-amplitude Variance for Various Beam TypesCollimated Beam:Θ0 = 1

Λ0 = 0 − 2α A(α) C̃2

nπ2k3−α/2Lα/2Γ(

2−α2

)

cos[

(α−2)π4

]

0≤ Λ0 < ∞ 1α−1A(α) C̃2


2−α2

)

{

(

Λ01+Λ2

0)

)α−2

21F1

(

2−α2 ;1; 2ρ2

W2

)

− 2α−2α

∞∑

n=0

( 2−α2 )n( α

2 )n

( 2+α2 )nn!

(

Λ20

1+Λ20)

)n/2cos

[

n tan−1(

1Λ0

)

+(α−2)π

4

]

}

Λ0 = ∞ − 2α A(α) C̃2

nπ2k3−α/2Lα/2 Γ( α2 )Γ( 2−α

2 )Γ( 2+α2 )

Γ(α) cos[

(α−2)π4

]

Divergent Beam and Convergent Beam: 0.5≤ Θ0

Λ0 = 0 − 2α A(α) C̃2

nπ2k3−α/2Lα/2Γ(2−α

2

)

2F1(2−α

2 , α2 ; 2+α

2 ;Θ̃)

cos[

(α−2)π4

]

0≤ Λ0 < ∞ 1α−1A(α) C̃2


2−α2

)

{

Λα−2

2 1F1

(

2−α2 ;1; 2ρ2

W 2

)

− 2α−2α

∞∑

n=0

( 2−α2 )n( α

2 )n

( 2+α2 )nn!

(Θ̃2 + Λ2)n/2cos[

n tan−1(

ΛΘ̃

)

+ (α−2)π4

]

}

Λ0 = ∞ − 2α A(α) C̃2

nπ2k3−α/2Lα/2 Γ( α2 )Γ( 2−α

2 )Γ( 2+α2 )

Γ(α)cos

[

(α−2)π4

]

Convergent Beam:Θ0 < 0.5, Θ0 6= 0

Λ0 = 0

− 2α A(α) C̃2

nπ2k3−α/2Lα/2Γ(2−α

2

)

Θα−2

2

×2F1(

2−α2 ,1; 2+α

2 ;1− 1Θ

)

cos[

(α−2)π4

]

, Θ0 > 0

−A(α) C̃2nπ2k3−α/2Lα/2

×{

2α Γ(1+ α/2)Γ(1−α)|Θ̃|−

α2 cos

[

(2−3α)π4

]

+ Γ(2−α

2

)

× 1α−1|Θ̃|

α−22 2F1

(

2−α2 ,1−α;2−α; 1

|Θ̃|

)

cos[

(α−2)π4

]}

, Θ0 < 0

0≤ Λ0 < ∞ 1α−1A(α) C̃2


2−α2

)

{

Λα−2

2 1F1

(

2−α2 ;1; 2ρ2

W 2

)

−(Θ̃2 + Λ2)α−2

4∞∑

n=0

( 2−α2 )n(1−α)n

(2−α)nn! (Θ̃2 + Λ2)−n/2cos[(

n− α−22

)

× tan−1( ΛΘ̃ )+ (α−2)π

4

]

− 2α−2α

Γ(1+α/2)Γ(1−α)Γ(1−α/2) (Θ̃2 + Λ2)−

α4

×cos[

α2 tan−1( Λ

Θ̃ )+ (2−3α)π4

]}

Λ0 = ∞ − 2α A(α) C̃2

nπ2k3−α/2Lα/2 Γ( α2 )Γ( 2−α

2 )Γ( 2+α2 )

Γ(α)cos

[

(α−2)π4

]

Perfect Focused Beam:Θ0 = 0

Λ0 = 0{

0, ρ = 0∞, ρ > 0

0≤ Λ0 < ∞ 1α−1A(α) C̃2


2−α2

)

{

Λ2−α

20 1F1

(

2−α2 ;1; 2ρ2

W 2

)

−(

1+Λ20

Λ20

)

α−24 ∞

∑n=0

( 2−α2 )n(1−α)n

(2−α)nn!

(

Λ20

1+Λ20

)n/2cos

[(

n− α−22

)

× tan−1(

1Λ0

)

+ (α−2)π4

]

− 2α−2α

Γ(1+α/2)Γ(1−α)Γ(1−α/2)

(

Λ20

1+Λ20

)

α4

×cos[

α2 tan−1

(

1Λ0

)

+ (2−3α)π4

]}

Λ0 = ∞ − 2α A(α) C̃2

nπ2k3−α/2Lα/2 Γ( α2 )Γ( 2−α

2 )Γ( 2+α2 )

Γ(α) cos[

(α−2)π4

]


3.1. Collimated beam

For a collimated beam(Θ0 = 1), based on Eqs. (5) and (14), the longitudinal component isexpressed as

σ2χ ,l(α) =

1α −1


(

2−α2

)

{

(

Λ0

1+ Λ20

)α−2

2

−2α −2

α

×∞

∑n=0

(2−α2 )n(

α2 )n

(2+α2 )nn!

(

Λ20

1+ Λ20)

)n/2

cos

[

n tan−1(

1Λ0

)

+(α −2)π

4

]

}

. (17)

Because the radial component disappears in the limiting case of a plane wave or a sphericalwave, the longitudinal component, Eq. (17), reduces to the plane-wave log-amplitude variancein the limiting caseΛ0 = 0 ,

σ2χ ,p(α) = −

2α


(

2−α2

)

cos

[

(α −2)π4

]

, (18)

while reducing to the spherical-wave one in the limiting caseΛ0 = ∞,

σ2χ ,s(α) = −

2α

A(α) C̃2nπ2k3−α/2Lα/2 Γ

( α2

)

Γ(2−α

2

)

Γ(2+α

2

)

Γ(α)cos

[

(α −2)π4

]

. (19)

Although Eqs. (18) and (19) are formally different from the results obtained by Beland [11],they are equivalent each other. In addition, whenα is set to 11/3, Eqs. (18) and (19) match theconventional Kolmogorov results for the plane and spherical waves perfectly [20].

The sum of longitudinal and radial components for a collimated beam is listed in Table 1. Inorder to analyze the influence of spectral power-law variations on the log-amplitude fluctuationsof a collimated beam, the log-amplitude variance is plottedin Fig. 1 as a function of power lawα and the Fresnel ratio at the transmitterΛ0 for several values of the ratioρ/W for a specialcase, takingL = 1000m; C̃2

n = 1.0×10−14m3−α ; λ = 1.55×10−6m. Figure 1 (a) depicts the on-axis log-amplitude fluctuations(ρ/W = 0), while Figure 1 (b) and 1(c) represent the off-axislog-amplitude fluctuations(ρ/W = 0.5,1.0). Since the path lengthL and optical wavelengthλ are fixed, all changes in the Fresnel ratio at the transmitterΛ0 = 2L/kW 2

0 correspond tovariations in the transmitter beam radiusW0.

As it is shown in Fig. 1(a), for some fixed Fresnel ratio for lower alpha values than 3.5 theon-axis log-amplitude variance increases up to a maximum value that occurs nearα = 3.2.At the maximum point the curve changes its slopes and the log-amplitude variance begins todecrease down to zero. In addition, for larger alpha values than 3.5 the log-amplitude varianceslightly decreases. These comments are similar to those on the scintillations for a plane orspherical wave in Ref. [15]. Figure 1(b) and 1(c) show that the above comments about the on-axis log-amplitude variance also adapt to the off-axis log-amplitude variance for small or largeΛ0, while the off-axis log-amplitude variance in the vicinityof Λ0 = 1 firstly increases up toa peak value, then decreases down to a trough, lastly begins to increase again rather that tendsto zero as Fig. 1(a). This is due to the variations in the ratioρ/W . In order to further analyzethe influence of the ratioρ/W on the log-amplitude fluctuations, the log-amplitude variance forseveral values of the Fresnel ratio at the transmitterΛ0 = 0.01,1.00,100 is plotted in Fig. 2 asa function ofα and the ratioρ/W for the same case as Fig. 1. It is deduced from Fig. 2 thatthe log-amplitude variance increases monotonically with the ratioρ/W for some fixed value ofalpha and increases slightly for small or large value ofΛ0 as shown in Fig. 2(a) and 2(c), whileincreasing sharply nearΛ0 = 1 as shown in Fig. 2(b). Furthermore, Figure 2(b) also shows that


10−2

100

102 3

3.5

40

0.05

0.1

0.15

αΛ

0

σ χ2 (ρ,α

)

10−2

100

102 3

3.5

4

0

0.05

0.1

0.15

αΛ0

σ χ2 (ρ,α

)

10−2

100

102 3

3.5

40

0.1

0.2

0.3

0.4

αΛ

0

σ χ2 (ρ,α

)

ρ/W=0.0 ρ/W=0.5 ρ/W=1.0

(a) (b) (c)

Fig. 1. The log-amplitude variance for a collimated beam(Θ0 = 1) as a function of powerlaw α and the Fresnel ratio at the transmitterΛ0 for several values of the ratioρ/W , with(a) for ρ/W = 0, (b) forρ/W = 0.5, and (c) forρ/W = 1.0.

00.5

1 33.5

40

0.02

0.04

0.06

0.08

0.1

0.12

αρ/W

σ χ2 (ρ,α

)

0

0.5

1 3

3.5

40

0.05

0.1

0.15

αρ/W

σ χ2 (ρ,α

)

00.5

1 33.5

0

0.02

0.04

0.06

0.08

αρ/W

σ χ2 (ρ,α

)

Λ0=100Λ

0=1.00Λ

0=0.05

(a) (b) (c)

Fig. 2. The log-amplitude variance for a collimated beam(Θ0 = 1) as a function of powerlaw α and the ratioρ/W for several values of the Fresnel ratio at the transmitterΛ0, with(a) for Λ0 = 0.05, (b) forΛ0 = 1.00, and (c) forΛ0 = 100.

for different values of alpha the log-amplitude variance inthe vicinity ofΛ0 = 1 increases withthe ratioρ/W with different speeds. Finally, it is emphasized that the radial component for acollimated beam disappears for allρ andα between the range 3 to 4 only asΛ0 = 0 andΛ0 = ∞.

3.2. Divergent beam

For a divergent beam(Θ0 > 1), using Eq. (14), the longitudinal component is given by

σ2χ , l(ρ ,α) =

1α −1


(

2−α2

)

×{

Λα−2

2

−2α −2

α

∞

∑n=0

(2−α2 )n(

α2 )n

(2+α2 )nn!

(Θ̃2 + Λ2)n/2cos

[

n tan−1(ΛΘ̃

)+(α −2)π

4

]

}

. (20)

The sum of longitudinal and radial components for a divergent beam is also listed in Table 1and are plotted in Figs. 3 and 4, respectively, in a similar manner to that for the collimated beamin Figs. 1 and 2 for the curvature parameter at the transmitter Θ0 = 2 for a special case. We takeL = 1000m; C̃2

n = 1.0×10−14m3−α ; λ = 1.55×10−6m. Figs. 3 and 4 are much like Figs. 1 and


10−2

100

102 3

3.5

4

0

0.02

0.04

0.06

0.08

0.1

αΛ0

σ χ2 (ρ,α

)

10−2

100

102 3

3.5

4

0

0.02

0.04

0.06

0.08

0.1

αΛ0

σ χ2 (ρ,α

)

10−2

100

102 3

3.2

3.4

3.6

3.8

4

0

0.1

0.2

αΛ0

σ χ2 (ρ,α

)

(a) (b) (c)

ρ/W=0.0 ρ/W=0.5 ρ/W=1.0

Fig. 3. The log-amplitude variance for a divergent beam(Θ0 = 2) as a function of powerlaw α and the Fresnel ratio at the transmitterΛ0 for several values of the ratioρ/W , with(a) for ρ/W = 0, (b) forρ/W = 0.5, and (c) forρ/W = 1.0.

00.5

1 33.5

4

0

0.02

0.04

0.06

0.08

0.1

0.12

αρ/W

σ χ2 (ρ,α

)

00.5

1 3

3.5

40

0.05

0.1

0.15

0.2

αρ/W

σ χ2 (ρ,α

)

0

0.5

1 33.5

40

0.02

0.04

0.06

0.08

αρ/W

σ χ2 (ρ,α

)

Λ0=0.05 Λ

0=1.00 Λ

0=100

(a) (b) (c)

Fig. 4. The log-amplitude variance for a divergent beam(Θ0 = 2) as a function of powerlaw α and the ratioρ/W for several values of the Fresnel ratio at the transmitterΛ0, with(a) for Λ0 = 0.05, (b) forΛ0 = 1.00, and (c) forΛ0 = 100.

2, respectively, thus the same comments as for the collimated beam are also deduced from Figs.3 and 4. In addition, the radial component for the divergent beam, like that for the collimatedbeam, diminishes for allρ andα whenΛ0 = 0 andΛ0 = ∞. The longitudinal component for thedivergent beam approaches the log-amplitude variance of the spherical wave asΛ0 → ∞, whilenot tending to the log-amplitude variance of the plane wave as Λ0 → 0 because the curvatureparameter at the receiverΘ for a divergent beam tends to the valueΘ = 1/Θ0 rather than unityas for a collimated beam. For this limiting caseΛ0 → 0, the longitudinal component, like theKolmogorov case, moves from the log-amplitude variance of the plane wave to that of thespherical wave asΘ0 increases from unity. Here it is noted that the log-amplitude variances forthe plane and spherical waves mentioned above are all the function of the power lawα.

3.3. Convergent beam

For a convergent beam(Θ0 < 1), whenΘ0 ≥ 0.5, the longitudinal component has the sameexpression as that of the divergent beam given by Eq. (20) above. But whenΘ0 < 0.5, using


Eq. (16) leads to

σ2χ ,l(ρ ,α) =

1α −1


(

2−α2

)

{

Λα−2

2 − (Θ̃2+ Λ2)α−2

4

×∞

∑n=0

(2−α2 )n(1−α)n

(2−α)nn!(Θ̃2 + Λ2)−n/2cos

[(

n−α −2

2

)

tan−1(ΛΘ̃

)

+(α −2)π

4

]

−2α −2

αΓ(1+ α/2)Γ(1−α)

Γ(1−α/2)(Θ̃2 + Λ2)−

α4

×cos

[

α2

tan−1(ΛΘ̃

)+(2−3α)π

4

]}

. (21)

Furthermore, for the traditional case of a perfectly focused beam characterized byΘ0 = 0, thelongitudinal component is given by

σ2χ ,l(ρ ,α) =

1α −1


(

2−α2

)

Λ2−α

20 −

(

1+ Λ20

Λ20

)

α−24

×∞

∑n=0

(2−α2 )n(1−α)n

(2−α)nn!

(

Λ20

1+ Λ20

)n/2

cos

[(

n−α −2

2

)

tan−1(

1Λ0

)

+(α −2)π

4

]

−2α −2

αΓ(1+ α/2)Γ(1−α)

Γ(1−α/2)

(

Λ20

1+ Λ20

)

α4

×cos

[

α2

tan−1(

1Λ0

)

+(2−3α)π

4

]}

. (22)

10−2

100

102 33.5

40

0.01

0.02

0.03

0.04

0.05

0.06

αΛ0

σ χ2 (ρ,α

)

10−2

100

102 33.5

40

5

10

15

αΛ0

σ χ2 (ρ,α

)

10−2

100

102 33.5

40

10

20

30

40

50

60

αΛ0

σ χ2 (ρ,α

)

ρ/W=0.5 ρ/W=1.0ρ/W=0.0

(a) (b) (c)

Fig. 5. The log-amplitude variance for a perfectly focus beam (Θ0 = 0) as a function ofpower lawα and the Fresnel ratio at the transmitterΛ0 for several values of the ratioρ/W ,with (a) forρ/W = 0, (b) forρ/W = 0.5, and (c) forρ/W = 1.0.

The expressions of the log-amplitude variance for the convergent beam and the perfectly fo-cused beam appear in Table 1. The log-amplitude variance forthe perfectly focused beam isplotted in Fig. 5 as a function of power lawα and the Fresnel ratio at the transmitterΛ0 for sev-eral values of the ratioρ/W for a special case, takingL = 1000m; C̃2

n = 1.0× 10−14m3−α ;λ = 1.55× 10−6m. Figure 5 is completely different from Figs. 1 and 3. The on-axis log-amplitude variance transmits smoothly from zero(Λ0 → 0) to the log-amplitude variance ofthe spherical wave(Λ0 → ∞). The off-axis log-amplitude variance for smallΛ0 becomes very


large and increases rapidly withα, while that for largeΛ0 also increases to a peak value andthen decreases up to zero, which are similar to the log-amplitude variance of the sphericalwave. This is because the longitudinal component for a perfectly focused beam tends to thelog-amplitude variance of the spherical wave and the radialcomponent disappears asΛ0 → ∞,whereas asΛ0 → 0 the longitudinal component disappears and the radial component becomesunbounded forρ 6= 0, while vanishing forρ = 0.

10−2

100

102 33.5

40

0.05

0.1

0.15

0.2

0.25

αΛ0

σ χ2 (ρ,α

)

10−2

100

102 33.5

40

0.05

0.1

0.15

0.2

0.25

0.3

αΛ

0

σ χ2 (ρ,α

)

10−2

100

102 33.5

40

0.05

0.1

0.15

0.2

αΛ0

σ χ2 (ρ,α

)

Θ0=0.01 Θ

0=0.10 Θ

0=0.50

(a) (b) (c)

Fig. 6. The on-axis log-amplitude variance as a function of power lawα and the Fresnelratio at the transmitterΛ0 for several curvature parameters at the transmitterΘ0, with (a)for Θ0 = 0.01, (b) forΘ0 = 0.10, and (c) forΘ0 = 0.50.

10−2100

1023

3.54

0

5

10

15

20

25

30

α

Λ0

σ χ2 (ρ,α

)

10−2100

1023

3.54

0

0.5

1

1.5

2

2.5

3

3.5

α

Λ0

σ χ2 (ρ,α

)

10−2100

1023

3.54

0

0.2

0.4

0.6

0.8

α

Λ0

σ χ2 (ρ,α

)

Θ0=0.01 Θ

0=0.10 Θ

0=0.50

(c)(b)(a)

Fig. 7. The diffractive edge(ρ/W = 1) log-amplitude variance as a function of power lawαand the Fresnel ratio at the transmitterΛ0 for several curvature parameters at the transmitterΘ0, with (a) forΘ0 = 0.01, (b) forΘ0 = 0.10, and (c) forΘ0 = 0.50.

In order to analyze the log-amplitude fluctuations for smallbut not zeroΘ0, the on-axis anddiffractive edge (ρ/W = 1) log-amplitude variances are plotted in Figs. 6 and 7 as a function ofpower lawα and the Fresnel ratio at the transmitterΛ0, respectively, for a special case, takingL = 1000m; C̃2

n = 1.0×10−14m3−α ; λ = 1.55×10−6m. Figure 6 shows that, whenΘ0 is smallbut not zero, the on-axis log-amplitude variance has a minimum value nearΛ0 = 1 and increasesrapidly asΛ0 approaches zero (Θ0 is the smaller, the rapider is the increase), eventually tendingto a nonzero limiting value. And it approaches the log-amplitude variance of the spherical waveasΛ0 tends to the unbounded. Furthermore, the on-axis log-amplitude variance for small butnot zeroΘ0 increases up to a peak value, and at the maximum point the curve changes slope and


decreases down to zero with the increase of the power lawα for some fixed Fresnel ratio, whichare similar to that for the collimated or divergent waves (Figure 1(a) or 3(a)). Figure 7(a) showsthat the diffractive edge log-amplitude variance forΘ0 = 0.01, similar to that for the perfectlyfocused beam (Figure 5(c)), also becomes very large for small Λ0 and increases rapidly withα,while that for largeΛ0 increases to a peak value and then decreases up to zero. Figure 7(b) and7(c) show that the diffractive edge log-amplitude variances for Θ0 = 0.1 andΘ0 = 0.5, similarto that for the collimated or divergent waves (Figure 1(c) or3(c)), increase to a peak value andthen decrease up to zero with increase of alpha for small or largeΛ0, while they in the vicinityof Λ0 = 1 firstly increase up to a peak value, then arrive at a trough, lastly begin to increaseagain. Finally, the radial component for a convergent beam,like the Kolmogorov case, tends tozero asΛ0 → 0 andΛ0 → ∞, while the longitudinal component approaches the log-amplitudevariance of the spherical wave forΛ0 → ∞ while tending to a limiting value forΛ0 → 0.

Finally, the series in main equations in Table 1 converge quite rapidly so that the errors aresmaller than 1 % when the first five terms are calculated. Therefore, the first five terms of theseries are used to calculated the log-amplitude variance inall of Figures in the paper.

4. Conclusion

In this paper, the log-amplitude variances for a Gaussian-beam wave propagating in non-Kolmogorov turbulence are derived in the weak-fluctuation regime for a horizonal path fol-lowing the procedure already used from Miller and Ricklin [19] and using a non-Kolmogorovtheoretical power spectrum, which has a generalized power law and a generalized amplitudefactor instead of the standard Kolmogorov power law 11/3 anda constant amplitude factor0.033. The analytical expressions are obtained and also summarized in Table 1, like the Kol-mogorov case, for convenient reference and comparison withthe conventional Kolmogorovresults. Here it is especially noted that the expressions developed match perfectly with the con-ventional Kolmogorov results [19], respectively, when thespectral power law is equal to thestandard Kolmogorov value 11/3.

Based on the expressions developed, the effect of spectral power-law variations on the log-amplitude fluctuations for Gaussian-beam wave is analyzed for a particular case. It can beconcluded that the on-axis log-amplitude variance and the off-axis log-amplitude variance forsmall or largeΛ0 for a collimated or divergent beam, similar to the scintillations for the plane orspherical waves, increase up to a peak value, and at the maximum point the curve change slopeand decrease down to zero with the increase of the power lawα for some fixed Fresnel ratio,but the off-axis log-amplitude variance in the vicinity ofΛ0 = 1 is completely different fromthe on-axis one owing to the variations of distance from the beam center line, it firstly increasesup to a peak value, then arrives at a trough, lastly begins to increase again. However, for aperfectly focused beam, the on-axis log-amplitude variance transmits smoothly from zero tothe log-amplitude variance of the spherical wave. The off-axis log-amplitude variance for smallΛ0 becomes very large and increases rapidly withα, while that for largeΛ0 also increases toa peak value and then decreases up to zero, which are similar to the log-amplitude variance ofthe spherical wave. Finally, for some fixed alpha value the influence of the Fresnel ratioΛ0 andthe ratioρ/W on the log-amplitude variance is also analyzed, it is shown that the influence ofthe Fresnel ratioΛ0 and the ratioρ/W on the log-amplitude variance is similar to that for theKolmogorov case.

Acknowledgement

This research was financially supported by the National Natural Science Foundation of China(NSFC)(No.10374023 and 60432040). The authors are grateful for a grant from NSFC.


log-amplitude variance for a gaussian-beam wave propagating through non-kolmogorov turbulence

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