Temporal-frequency spectra for optical wave propagating through non-Kolmogorov turbulence

Temporal-frequency spectra for opticalwave propagating through

non-Kolmogorov turbulence

Wenhe Du1,2∗, Liying Tan1, Jing Ma1, and Yijun Jiang1,1National Key Laboratory of Tunable Laser Technology, Harbin Institute of Technology,

Harbin 150001, China2College of Science, Qiqihar University, Qiqihar 161006, China

[email protected]

Abstract: Nowadays it has been accepted that the Kolmogorov modelis not the only possible turbulent one in the atmosphere, which has beenconfirmed by the increasing experimental evidences and some results oftheoretical investigation. This has prompted the scientist community tostudy optical propagation in non-Kolmogorov atmospheric turbulence. Inthis paper, using a non-Kolmogorov power spectrum which has a more gen-eral power law instead of standard Kolmogorov power law value 11/3 and amore general amplitude factor instead of constant value 0.033, the temporalpower spectra of the presentative amplitude and phase effects, irradianceand angle of arrival fluctuations, have been derived for horizontal link inweak turbulence. And then the influence of spectral power-law variations onthe temporal power spectrum has been analyzed. It is anticipated that thiswork is helpful to the investigations of atmospheric turbulence and opticalwave propagation in the atmospheric turbulence.

© 2010 Optical Society of America

OCIS codes: (010.1290) Atmospheric optics; (010.1330) Atmospheric turbulence; (290.5930)Scintillation; (010.7350) Angle of arrival; (070.4790) Temporal power spectrum.

References and links1. J. C. Ricklin, S. M. Hammel, F. D. Eaton, and S. L. Lachinova, “Atmospheric channel effects on free-space laser

communication,” J. Opt. Fiber. Commun. Rep. 3, 111-158 (2006).2. K. Kazaura, K. Omae, T. Suzuki, and M. Matsumoto, “Enhancing performance of next generation FSO commu-

nication systems using soft computing-based predictions,” Opt. Express 14, 4958-4968 (2006).3. L. C. Andrews, R. L. Phillips, and P. T. Yu, “ Optical scintillations and fade statistics for a satellite-communication

system,” Appl. Opt. 34, 7742-7751 (1995).4. M. S. Be1en’kii, S. J. Karis, J. M. Brown, and R. Q. Fugate, “Experimental study of the effect of non-Kolmogorov

stratospheric turbulence on star image motion,” Proc. SPIE 3126, 113-123 (1997).5. D. T. Kyrazis, J. Wissler, D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in

the upper troposphere and lower stratosphere,” Proc. SPIE 2110, 43-55 (1994).6. G. Wang, “A new random-phase-screen time series simulation algorithm for dynamically atmospheric turbulence

wave-front generator,” Proc. SPIE 6027, 602716-1-12 (2006).7. A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol

turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence,” Atmospheric Re-search 88, 66-77 (2008).

8. M. S. Belen’kii, E. Cuellar, K. A. Hughes, and V. A. Rye “Experimental study of spatial structure of turbulenceat Maui Space Surveillance Site (MSSS),” Proc. SPIE 6304, 63040U-1-12 (2006).

9. 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).

(C) 2010 OSA 15 March 2010 / Vol. 18, No. 6 / OPTICS EXPRESS 5763#121049 - $15.00 USD Received 7 Dec 2009; revised 31 Jan 2010; accepted 5 Feb 2010; published 8 Mar 2010

10. E. Golbraikh and N. S. Kopeika, “Behavior of structure function of refraction coefficients in different trubulentfields,” Appl. Opt. 43, 6151-6156 (2004).

11. S. S. Moiseev and O. G. Chkhetiani, “Helical scaling in turbulence,” JETP 83, 192-198 (1996).12. T. Elperin, N. Kleeorin, and I. Rogachevskii, “Isotropic and anisotropic spectra of passive scalar fluctuations in

turbulent fluid flow,” Phys. Rev. E 53, 3431-3441 (1996).13. R. R. Beland, “Some aspects of propagation through weak isotropic non-Kolmogorov turbulence,” Proc. SPIE

2375, 1111-1126 (1995).14. B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric

turbulence,” Proc. SPIE 2471, 181-196 (1995).15. G. D. Boreman and C. Dainty, “Zernike expansions for non-Kolmogorov turbulence,” J. Opt. Soc. Am. A 13,

517-522 (1996).16. A. S. Gurvich and M. S. Belen’kii, “Influence of stratospheric turbulence on infrared imaging,” J. Opt. Soc. Am.

A 12, 2517-2522 (1995).17. M. S. Belen’kii, “Effect of the stratosphere on star image motion,” Opt. Lett. 20, 1359-1361 (1995).18. 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).19. C. Rao, W. Jiang, and N. Ling, “Spatial and temporal characterization of phase fluctuations in non-Kolmogorov

atmospheric turbulence,” J. Mod. Opt. 47, 1111-1126 (2000).20. V. I. Tatarski, Wave Propagating in a Turbulent Medium, (McGraw-Hill, New York, 1961).21. R. Conan, J. Borgnino, A. Ziad, and F. Martin, “Analytical solution for the covariance and for the decorrelation

time of the angle of arrival of a wave front corrugated by atmospheric turbulence,” J. Opt. Soc. Am. A 17,1807-1818 (2000).

22. D. M. Winker, “Effect of a finite outer scale on the Zernike decomposition of atmospheric optical turbulence,” J.Opt. Soc. Am. A 11, 1568-1574 (1991).

23. P. Hickson, “Wave-front curvature sensing from a single defocused image,” J. Opt. Soc. Am. 11, 1667-1673(1994) .

24. A. Erdelyi, Tables of Integral Transforms, (McGraw-Hill, New York, 1959).25. C. Ho and A. Wheelon, Power Spectrum of Atmospheric Scintillation for the Deep Space Network Goldstone

Ka-band Downlink, (Jet Propulsion Laboratory, California, 2004).26. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, (Dover, New York, 1965).27. A. Ishimaru, Wave Propagation and Scattering in Random Media, (Academic Press, New York, 1978).28. W. L. Wolf and G. J. Zissis, The Infrared Handbook, (Office of Naval Research, Washington, 1978).29. W. Du, S. Yu, L. Tan, J. Ma, Y. Jiang, and W. Xie, “Angle-of-arrival fluctuations for wave propagation through

non-Kolmogorov turbulence,” Opt. Commun. 282, 705-708 (2009).30. Gao Chong, Ma Jing, and Tan Liying, “Angle-of-arrival fluctuation of light beam propagation in strong turbulence

regime,” High Power Laser and Particle Beams 18, 891-894 (2006).31. L. C. Andrews and R.L. Phillips, Laser Propagation through Random Media, (SPIE Optical Engineering Press,

Bellingham, 1998).32. S. F. Clifford, “Temporal-frequency spectra for a spherica wave propagating through atmospheirc turbulence,” J.

Opt. Soc. Am. 61, 1285-1292 (1971).

1. Introduction

It is well-known that atmospheric turbulence severely degrades the performance of imaging andlaser systems [1, 2, 3]. For a long time, the Kolmogorov model for atmospheric turbulence hasbeen widely applied to estimate the performance of imaging and laser systems operating in theatmosphere, which has been confirmed by numerous experimental evidences.

Despite the success of the Kolmogorov model, recently both the experimental data [4, 5, 6,7, 8] and the theoretical investigations [9, 10, 11, 12] have shown that it is not the only possibleturbulent one in the atmosphere. This has prompted the scientist community to research opticalpropagation in non-Kolmogorov atmosphere turbulence. Beland developed the expressions oflog-amplitude variance and the coherence length for optical wave propagating through weakisotropic non-Kolmogorov turbulence [13]. Stribling et al analyzed the wave structure functionand the Strehl ratio as the refractive-index fluctuations deviated from Kolmogorov statistics[14]. Boreman and Dainty investigated the expressions of non-Kolmogorov turbulence in termsof Zernike polynomials [15]. Gurvich and Belen’kii presented a model for the power spectrumof stratospheric non-Kolmogorov turbulence and researched the influence of the stratosphericturbulence on the scintillation and the coherence of starlight and on the degradation of star


image [16]. And then the effect of the stratosphere on star image motion was analyzed againbased on the model for the power spectrum of stratosphere [17]. Tosellia et al introduced anon-Kolmogorov theoretical power spectrum model and analyzed long term beam spread, scin-tillation index, probability of fade, mean SNR, and mean BER as variations of the spectrumexponent for horizontal link [18]. Rao et al analyzed the spatial and temporal characterizationsof phase fluctuations in non-Kolmogorov atmospheric turbulence using a theoretical method[19]. The spatial structure function, the temporal structure function, and the temporal powerspectrum of phase fluctuations were derived. However, the temporal characteristics for the pre-sentative amplitude and phase effects, irradiance and angle-of-arrival (AOA) fluctuations, werenot discussed in their paper.

In this paper, we consider a non-Kolmogorov theoretical power spectrum for the refractive-index fluctuations [18], which obeys a more general power law that takes all the values betweenthe range 3 to 4. When the power law is set to the standard Kolmogorov value 11/3, the spectrumreduces to the conventional Kolmogorov one. Using this spectrum, the temporal power spectraof irradiance and AOA fluctuations have been developed for horizontal link in weak turbulence,and then the effect of spectral power-law variations on the temporal power spectrum has beenanalyzed.

2. Non-Kolmogorov spectrum

In order to research the temporal power spectra of irradiance and AOA fluctuations for opticalwave propagating in non-Kolmogorov atmospheric turbulence, a theoretical power spectrummodel for the refractive-index fluctuations [18] is considered, which obeys an arbitrary powerlaw and in which the power-law exponents can assume all the values ranging from 3 to 4,

Φn(κ,α) = A(α) C2n κ−α , 2π/L0 � κ � 2π/l0, 3 < α < 4, (1)

where κ denotes the magnitude of the spatial-frequency vector with units rad/m, α is thespectral power-law exponent, C2

n represents a generalized refractive-index structure parameterwith units m3−α , which describes the strength of turbulence along the link, l0 and L0 are theinner and outer scales of turbulence, respectively, and A(α) is defined by

A(α) =1

4π2 Γ(α −1) cos(απ

2

), (2)

and the symbol Γ(x) in the above expression denotes the gamma function. At α = 11/3,the function A(11/3) = 0.033 and C2

n=C2n , and the spectrum reduces to the conventional Kol-

mogorov spectrum,Φn(κ) = 0.033 C2

n κ−11/3, (3)

where C2n represents the conventional refractive-index structure parameter and has units of

m−2/3. In addition, as α approaches 3, A(α) approaches zero. As a result, the power spec-trum for refractive-index fluctuations vanishes in the limiting case α = 3. Finally, it can beseen from Eq. (1) that all of the analyses performed in this paper are only related to the inertialinterval of turbulent spectrum, i.e., 2π/L0 � κ � 2π/l0.

3. Temporal power spectrum of angle-of-arrival fluctuations

AOA fluctuations of an optical wave in the plane of receiver aperture are related to imagedancing in the focal plane of an imaging or laser systems. Therefore, it is very necessary toresearch the temporal power spectrum of AOA fluctuations.


According to the definition [20], the temporal power spectrum of AOA fluctuations Wθ (ω,β )is the Fourier transforms of the temporal covariance function of AOA Cθ (t,β ):

Wθ (ω,β ) = 4∫ ∞

0Cθ (t,β )cos(ωt)dt. (4)

The temporal covariance function in Eq. (4) may be determined using the Taylor frozen turbu-lence hypothesis, given the spatial covariance function.

The spatial covariance function of AOA for an optical wave propagating in the atmosphericturbulence is given by [21]

Cθ (ρ,β ) = πk−2∫ ∞

0κ3Wφ (κ)GD(κ)[J0(ρκ)− cos(2β )J2(ρκ)]dκ, (5)

where ρ represents the geometrical separation between points in the plane transverse to thedirection of propagation, β is the angle between the baseline and the AOA observation axis (seeFig. 1), k = 2π/λ and λ denotes the optical wavelength, Wφ (κ) is the wave-front phase powerspectrum, GD(κ) represents the point-spread function of the receiver aperture, and J0(ρκ) andJ2(ρκ) denote the zero and second order Bessel functions, respectively.

r

rv

x

y

Transverse wind

Fig. 1. Schematic layout of the baseline �ρ , the transverse wind velocity�v, the observationaxis x of the AOA fluctuations, and the angle β between �ρ and x axis.

Based on Eq. (5), the Taylor frozen turbulence hypothesis allows us to make the associa-tion �ρ =�vt, where �v denotes the transverse wind velocity. In the case the temporal covariancefunction of AOA can be written as

Cθ (t,β ) = πk−2∫ ∞

0κ3Wφ (κ)GD(κ)[J0(vtκ)− cos(2β )J2(vtκ)]dκ. (6)

3.1. Temporal power spectrum of AOA fluctuations for a plane wave

For a plane wave that propagates along the z axis from z = 0 to a receiver at z = L, the phasepower spectrum [22, 23] is given by

Wφ(pl)(κ) = 2πk2∫ L

0Φn(κ)cos2

(κ2z2k

)dz, (7)


where Φn(κ) is the power spectrum for the refractive-index fluctuations.For non-Kolmogorov turbulence, Eq. (7) can be written as

Wφ(pl)(κ,α) = 2πk2∫ L

0Φn(κ,α)cos2

(κ2z2k

)dz. (8)

Substituting Eq. (8) into Eq. (6) yields the temporal covariance function of AOA for a planewave

Cθ(pl)(α, t,β ) = 2π2∫ ∞

0κ3Φn(κ,α)GD(κ)[J0(vtκ)− cos(2β )J2(vtκ)]

×∫ L

0cos2

(κ2z2k

)dzdκ. (9)

Inserting Eq. (9) into Eq. (4), the temporal power spectrum of AOA fluctuations for a planewave is obtained,

Wθ(pl)(α,ω,β ) = 8π2∫ ∞


×∫ L

0cos2

(κ2z2k

)∫ ∞

0cos(ωt)dκdzdt. (10)

Using the integral relations [24],

∫ ∞

0J0(ax)cos(bx)dx =

{(a2 −b2)−1/2, 0 < b < a,0, b > a,

(11)

and∫ ∞

0J2n(ax)cos(bx)dx =

{(−1)n(a2 −b2)−1/2T2n(b/a), 0 < b < a,0, b > a,

(12)

where T2n(z) is Tchebichef polynomials [24],

Tm(z) = cos(mcos−1 z), (13)

the integrals with respect to t are evaluated. As a result, Eq. (10) is expressed as

Wθ(pl)(α,ω,β ) = 8π2∫ L

0dz∫ ∞

ω/vdκκ3Φn(κ,α)GD(κ)cos2

(κ2z2k

)

×{[

(vκ)2 −ω2]−1/2+ cos(2β )

[(vκ)2 −ω2]−1/2

×[2( ω

vκ

)2−1

]}. (14)

For typical imaging and laser systems, the receiver is usually a telescope with a pupil ofdiameter D. Its point-spread function can be modelled as the Gaussian function [25],

GD(κ) ≈ exp

(−c2D2κ2

4

), (15)

here c = 0.4832.


Substituting Eqs. (1) and (15) into Eq. (14) yields

Wθ(pl)(α,ω,β ) = 8π2A(α)C2n

∫ L

0dz∫ ∞

ω/vdκκ3−α exp

(−c2D2κ2

4

)

×cos2(

κ2z2k

){[(vκ)2 −ω2]−1/2

+ cos(2β )

×[(vκ)2 −ω2]−1/2[2( ω

vκ

)2−1

]}. (16)

Using the geometrical optics approximations (cos2(

κ2z2k

)= 1), as its condition that the Fresnel

zone (L/k)1/2 is much smaller than the receiver aperture diameter, (L/k)1/2 � D, is satisfied,and considering the integral relation [24]

∫ ∞

0(t +a)2μ−1(t −b)2ν−1 exp(−pt)dt

={

0, 0 < t < b,

Γ(2ν)(a+b)μ+ν−1 p−μ−νep(a−b)/2Wμ−ν ,μ+ν−1/2(bp+ap), t > b,(17)

the temporal power spectrum of AOA fluctuations for a plane wave propagating in non-Kolmogorov atmospheric turbulence is given by

Wθ(pl)(α,ω,β ) =7.09π2A(α)C2

nL1+α

4 k3−α

4

ω0

(ωω0

) 1−α2(

c2D2

4

) α−54

×exp

[−kc2D2

8L

(ωω0

)2]{

[1− cos(2β )]W3−α4 , 3−α

4

[kc2D2

4L

(ωω0

)2]

+2cos(2β )ωω0

(kc2D2

4L

) 12

W1−α4 , 1−α

4

[kc2D2

4L

(ωω0

)2]⎫⎬⎭ , (18)

where Wμ,ν(z) is Whittaker’s confluent hypergeometric function [26] and ω0 = v(L/k)−1/2.

3.2. Temporal power spectrum of AOA fluctuations for a spherical wave

For a spherical wave, the phase power spectrum [27, 28] is given by

Wφ(sp)(κ) = 2πk2∫ L

0Φn(κ)

( zL

)2cos2

[κ2z(L− z)

2kL

]dz. (19)

For non-Kolmogorov turbulence, Eq. (19) can be expressed as

Wφ(sp)(κ,α) = 2πk2∫ L

0Φn(κ,α)

( zL

)2cos2

[κ2z(L− z)

2kL

]dz. (20)

Substituting Eq. (20) into Eq. (6) yields the temporal covariance function of AOA for aspherical wave

Cθ(sp)(α, t,β ) = 2π2∫ ∞


×∫ L

0cos2

[κ2z(L− z)

2kL

]( zL

)2dzdκ. (21)


Following the same procedure as used above, the temporal power spectrum of AOA fluctu-ations for a spherical wave propagating in non-Kolmogorov atmospheric turbulence is givenby

Wθ(sp)(α,ω,β ) =2.36π2A(α)C2

nL1+α

4 k3−α

4

ω0

(ωω0

) 1−α2(

c2D2

4

) α−54

×exp

[−kc2D2

8L

(ωω0

)2]{

[1− cos(2β )]W3−α4 , 3−α

4

[kc2D2

4L

(ωω0

)2]

+2cos(2β )ωω0

(kc2D2

4L

) 12

W1−α4 , 1−α

4

[kc2D2

4L

(ωω0

)2]⎫⎬⎭ . (22)

0.01 0.1 1

10-10

10-5

100

W(p

l)(,

,)/

(pl)

2

/0

=3.1=3.3=3.5=3.7=3.9

0.01 0.1 1

10-10

10-5

100

/0

W(s

p)(

,,

)/(s

p)2

=3.1=3.3=3.5=3.7=3.9

(a) (b)

Fig. 2. The temporal power spectrum of AOA fluctuations scaled by the correspondingvariance as a function of the frequency ratio ω/ω0 for several values of power law α , with(a) for a plane wave, (b) for a spherical wave.

In the development of the temporal power spectra of AOA fluctuations, Eqs. (18) and (22),the geometrical optics approximation is used, thus the range of validity of Eqs. (18) and (22) is(L/k)1/2 � D. Moreover, the valid range of the power spectrum of the refractive-index fluctu-ations imposes the constraints l0 � (L/k)1/2 � L0, L0 � D, and ω > 2πv/L0 on Eqs. (18) and(22) again.

The temporal power spectra of AOA fluctuations for the plane and spherical waves, normal-ized to the appropriate AOA variance [29], are plotted as a function of the frequency ratio ω/ω0

in Fig. 2 for several values of power law α for a particular case, taking D = 0.25m and β = 0.Observe that the temporal spectrum for a plane wave is similar to that for a spherical wave.The temporal spectrum decays slightly with the frequency for ω<0.1ω0 and decays rapidly forω>0.1ω0. In addition, for ω<0.1ω0, the slopes of the curves decrease from 0 to -1 with therising of power law α , while for ω>0.1ω0 they increase from -4 to -3 with the rising of powerlaw α . Here it is noted that the power spectra for AOA fluctuations given by Eqs. (18) and(22) are consistent with the conventional results that corresponds to Kolmogorov atmospheric


0.01 0.1 1

10-10

10-5

100

/0

W(p

l)(,

,)/

(pl)

2

D=0.2mD=0.4mD=0.6mD=0.8mD=1.0m

0.01 0.1 1

10-10

10-5

100

/0

W(s

p)(

,,

)/(s

p)2

D=0.2mD=0.4mD=0.6mD=0.8mD=1.0m

(a) (b)

Fig. 3. The temporal power spectrum of AOA fluctuations scaled by the correspondingvariance as a function of the frequency ratio ω/ω0 for several values of the receiver apertureD, with (a) for a plane wave, (b) for a spherical wave.

0.01 0.1 110-15

10-10

10-5

100

/0

W(p

l)(,

,)/

(pl)

2

=0= /8= /4=3 /8= /2

0.01 0.1 110-15

10-10

10-5

100

/0

W(s

p)(

,,

)/(s

p)2

=0= /8= /4=3 /8= /2

(a) (b)

Fig. 4. The temporal power spectrum of AOA fluctuations scaled by the correspondingvariance as a function of the frequency ratio ω/ω0 for several values of β , with (a) for aplane wave, (b) for a spherical wave.


turbulence [30] when α is set to 11/3, although Eqs. (18) and (22) is not formally in agreementwith the conventional Kolmogorov results.

In addition, in order to analyze the effect of the receiver aperture D on the temporal spectrum,the normalized temporal spectra for the plane and spherical waves are plotted as a function ofthe frequency ratio ω/ω0 in Fig. 3 for several values of the receiver aperture D for a particularcase. We take α = 11/3 and β = 0. As it is shown in Fig. 3, with the increase of the receiveraperture D, the slopes of the curves decrease in the low-frequency band, while they are al-most the same in the high-frequency band. Moreover, it is also shown in Fig. 3 that increasingthe receiver aperture can filter higher frequencies, which come directly from the filter functionexp[−kc2D2ω2/(8Lω2

0 )]

in Eqs. (18) and (22). To research the effect of the observation orien-tation β on the temporal spectrum, also the normalized temporal spectra of AOA fluctuationsfor the plane and spherical waves are plotted as a function of the frequency ratio ω/ω0 in Fig.4 for several values of β for a particular case, taking α = 11/3 and D = 0.25m. As it is shownin Fig. 4, for ω<0.1ω0, the slopes of the curves decrease with the increase of beta, while forω>0.1ω0 they increase with the increase of beta. Finally, the comparison of the temporal powerspectrum for a plane wave with that for a spherical wave is shown in Fig. 5 for a special case,taking α = 11/3; D = 0.25m; β = 0. The result shows that the normalized temporal spectrumof AOA fluctuations for a plane wave is the same as that for a spherical wave when the powerlaw α is set to some value.

0.01 0.1 1

10-10

10-5

100

/0

W(

,,

)/2

Spherical wavePlane wave

Fig. 5. The temporal power spectrum of AOA fluctuations scaled by the correspondingvariance as a function of the frequency ratio ω/ω0 for power law α = 11/3. The solidcurve represents a spherical wave. The circle denotes a plane wave.

4. Temporal power spectrum of irradiance fluctuations

For free-space laser optics communications or laser radar systems, the irradiance fluctuationsresulting from the propagation of laser beam through the atmospheric turbulence is one of themain noise. Thus it is also very necessary to research the temporal power spectrum of irradiancefluctuations.

As for the case of AOA fluctuations, the temporal power spectrum of irradiance fluctuations


WI(ω,L) is related to the Fourier transform of the temporal covariance function of irradianceCI(t,L) by

WI(ω,L) = 4∫ ∞

0CI(t,L)cos(ωt)dt. (23)

4.1. Temporal power spectrum of irradiance fluctuations for a plane wave

The temporal covariance function of irradiance for a plane wave [31] is given by

CI(pl)(t,L) = 8π2k2∫ L

0

∫ ∞

0κΦn(κ)J0(vtκ)

[1− cos

(κ2(L− z)

k

)]dκdz. (24)

For non-Kolmogorov turbulence, Eq. (24) can be expressed as

CI(pl)(α, t,L) = 8π2k2∫ L

0

∫ ∞

0κΦn(κ,α)J0(vtκ)

[1− cos

(κ2(L− z)

k

)]dκdz. (25)

Substituting Eq. (25) into Eq. (23) yields the temporal power spectrum of irradiance for aplane wave,

WI(pl)(α,ω,L) = 32π2k2∫ L

0dz∫ ∞

0dκ∫ ∞


×[1− cos

(κ2(L− z)

k

)]cos(ωt)dt. (26)

Following the same procedure already used from Clifford [32], the temporal power spec-trum of irradiance fluctuations for a plane wave propagating in non-Kolmogorov atmosphericturbulence is given by

WI(pl)(α,ω,L) =16π2k

6−α2 A(α)C2

nLα2

ω0

(ωω0

)1−α Γ(

12

)Γ(α−1

2

)

Γ(α

2

) Re{1

− 2F2

(1,

2−α2

;32,3−α

2; i

12

(ωω0

)2)− Γ

(α2

)Γ(

1−α2

)Γ(

1+α2

)

Γ(α−1

2

)Γ(

3+α2

)

×(−i

12

(ωω0

)2) α−1

2

1F1

(12

;α +3

2; i

12

(ωω0

)2)⎫⎬⎭ . (27)

where 1F1(a;c;z) is the confluent hypergeometric function of the first kind and2F2(a1,a2;c1,c2;z) is the generalized hypergeometric function.

4.2. Temporal power spectrum of irradiance fluctuations for a spherical wave

The temporal covariance function of irradiance for a spherical wave [31] is given by

CI(sp)(t,L) = 8π2k2∫ L

0

∫ ∞

0κΦn(κ)J0(vtκ)

[1− cos

(κ2z(L− z)

Lk

)]dκdz. (28)

For non-Kolmogorov turbulence, the above formula can be written as

CI(sp)(α, t,L) = 8π2k2∫ L

0

∫ ∞


[1− cos

(κ2z(L− z)

Lk

)]dκdz. (29)


Substituting Eq. (29) into Eq. (23) yields the temporal power spectrum of irradiance fluctua-tions for a spherical wave,

WI(sp)(α,ω,L) = 32π2k2∫ L

0dz∫ ∞

0dκ∫ ∞


×[1− cos

(κ2z(L− z)

Lk

)]cos(ωt)dt. (30)

1000

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

/0

WI(

pl)(

)/I(

pl)

2

=3.1=3.3=3.5=3.7=3.9

100

2

4

6

8

10

12

14

x 10-3

/0

WI(

sp)(

)/I(

sp)

2=3.1=3.3=3.5=3.7=3.9

(a) (b)

Fig. 6. The temporal power spectrum of irradiance fluctuations scaled by the correspondingvariance as a function of the frequency ratio ω/ω0 for several values of power law α , with(a) for a plane wave, (b) for a spherical wave.

Following the same procedure as used above, the temporal power spectrum of irradiancefluctuations for a spherical wave propagating in non-Kolmogorov atmospheric turbulence isgiven by

WI(sp)(α,ω,L) =16π2k

6−α2 A(α)C2

nLα2

ω0

(ωω0

)1−α Γ(

12

)Γ(α−1

2

)

Γ(α

2

) Re{1

− 2F2

(1,

2−α2

;32,3−α

2; i

14

(ωω0

)2)− 1

2

Γ(α

2

)Γ(

1−α2

)Γ(

1+α2

)

Γ(α−1

2

)Γ(

2+α2

)

×(−i

14

(ωω0

)2) α−1

2

1F1

(12

;α +2

2; i

14

(ωω0

)2)⎫⎬⎭ . (31)

As for the case of AOA fluctuations, the temporal power spectra of irradiance fluctuations,Eqs. (27) and (31), are also subject to the constraints l0 � (L/k)1/2 � L0 and ω > 2πv/L0. Inaddition, since we are mainly concerned with the influence of the variations of spectral powerlaw α on the temporal power spectrum of irradiance fluctuations here, the receiver aperture Dis not included in Eqs. (27) and (31). Finally, it is noteworthy that, when the spectral power law


0.01 0.01 10

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

/0

WI(

)/I2

Plane wave

Spherical wave

Fig. 7. The temporal power spectrum of irradiance fluctuations scaled by the correspondingvariance as a function of the frequency ratio ω/ω0 for power law α = 11/3. The dashedcurve represents a plane wave, the dotted curve represents a spherical wave.

α is set to 11/3, Eqs. (27) and (31) match the conventional Kolmogorov results for the planeand spherical waves perfectly.

The temporal power spectra of irradiance fluctuations for the plane and spherical waves,normalized to the appropriate irradiance variance [13], are plotted as a function of the frequencyratio ω/ω0 in Fig. 6 for several values of power law α . Observe that the temporal spectrum for aplane wave is similar to that for a spherical wave. The temporal spectrum is essentially constantfor ω < ω0, while decaying as ω1−α for ω > ω0. In addition, the temporal spectrum increaseswith the rising of power law α . The comparison of the temporal power spectrum of a planewave with that of a spherical wave is shown in Fig. 7 for power law α = 11/3. It is shown thatthe temporal power spectrum for a spherical wave is smaller than that for a plane wave and thatthe spherical wave spectrum extends to higher frequencies.

5. Conclusion

In this paper, the temporal power spectra of the irradiance and AOA fluctuations of a planeand spherical waves are derived for horizontal link in weak turbulence using a generalizedpower law spectrum which owns a generalized power law and in which the power-law exponentvaries from 3 to 4. It is noteworthy that all of the expressions of the temporal power spectrumdeveloped here are analytical.The derived expressions are used to analyze the effect of spectralpower-law variations on the temporal power spectrum.

The results show that the temporal power spectrum of AOA fluctuations for a plane waveis similar to that for a spherical wave. It decays slightly with the frequency for ω<0.1ω0 anddecays rapidly for ω>0.1ω0. In addition, for ω<0.1ω0, the slopes of the curves of the nor-malized temporal spectrum versus ω/ω0 decrease with the rising of power law α , while forω>0.1ω0 they increase. The temporal spectrum of irradiance fluctuations for a plane wave isalso similar to that for a spherical wave. It is essentially constant for ω < ω0, while decayingas ω1−α for ω > ω0. And the temporal spectrum increases with the rising of power law α . It


is also shown that the temporal power spectrum for a spherical wave is smaller than that for aplane wave and that the spherical wave spectrum extends to higher frequencies for some valueof alpha. The results will contribute to the investigations of atmospheric turbulence and opticalwave propagation in the atmospheric turbulence.

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.


Documents

Temporal-frequency spectra for optical wave propagating through non-Kolmogorov turbulence