Abstract

Nowadays it has been accepted that the Kolmogorov model is not the only possible turbulent one in the atmosphere, which has been confirmed by the increasing experimental evidences and some results of theoretical investigation. This has prompted the scientist community to study optical propagation in non-Kolmogorov atmospheric turbulence. In this paper, using a non-Kolmogorov power spectrum which has a more general power law instead of standard Kolmogorov power law value 11/3 and a more general amplitude factor instead of constant value 0.033, the temporal power spectra of the presentative amplitude and phase effects, irradiance and angle of arrival fluctuations, have been derived for horizontal link in weak turbulence. And then the influence of spectral power-law variations on the temporal power spectrum has been analyzed. It is anticipated that this work is helpful to the investigations of atmospheric turbulence and optical wave propagation in the atmospheric turbulence.

© 2010 Optical Society of America

1. Introduction

It is well-known that atmospheric turbulence severely degrades the performance of imaging and laser systems [1, 2, 3]. For a long time, the Kolmogorov model for atmospheric turbulence has been widely applied to estimate the performance of imaging and laser systems operating in the atmosphere, 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 possible turbulent one in the atmosphere. This has prompted the scientist community to research optical propagation in non-Kolmogorov atmosphere turbulence. Beland developed the expressions of log-amplitude variance and the coherence length for optical wave propagating through weak isotropic non-Kolmogorov turbulence [13]. Stribling et al analyzed the wave structure function and the Strehl ratio as the refractive-index fluctuations deviated from Kolmogorov statistics [14]. Boreman and Dainty investigated the expressions of non-Kolmogorov turbulence in terms of Zernike polynomials [15]. Gurvich and Belen’kii presented a model for the power spectrum of stratospheric non-Kolmogorov turbulence and researched the influence of the stratospheric turbulence 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 again based on the model for the power spectrum of stratosphere [17]. Tosellia et al introduced a non-Kolmogorov theoretical power spectrum model and analyzed long term beam spread, scintillation index, probability of fade, mean SNR, and mean BER as variations of the spectrum exponent for horizontal link [18]. Rao et al analyzed the spatial and temporal characterizations of phase fluctuations in non-Kolmogorov atmospheric turbulence using a theoretical method [19]. The spatial structure function, the temporal structure function, and the temporal power spectrum of phase fluctuations were derived. However, the temporal characteristics for the presentative amplitude and phase effects, irradiance and angle-of-arrival (AOA) fluctuations, were not 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 between the range 3 to 4. When the power law is set to the standard Kolmogorov value 11/3, the spectrum reduces to the conventional Kolmogorov one. Using this spectrum, the temporal power spectra of 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 been analyzed.

2. Non-Kolmogorov spectrum

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

Φnκα=A(α)C˜n2κα,2π/L0κ2π/l0,3<α<4,

where κ denotes the magnitude of the spatial-frequency vector with units rad/m, α is the spectral power-law exponent, 2n represents a generalized refractive-index structure parameter with units m3-α, which describes the strength of turbulence along the link, l0 and L0 are the A(α) is defined by

A(α)=14π2Γ(α1)cos(απ2),

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 2n=C2n, and the spectrum reduces to the conventional Kolmogorov spectrum,

Φn(κ)=0.033Cn2κ11/3,

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 spectrum for refractive-index fluctuations vanishes in the limiting case α = 3. Finally, it can be seen from Eq. (1) that all of the analyses performed in this paper are only related to the inertial interval of turbulent spectrum, i.e., 2π/L0κ ≪ 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 image dancing in the focal plane of an imaging or laser systems. Therefore, it is very necessary to research 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):

Wθωβ=40Cθtβcos(ωt)dt.

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

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

Cθρβ=πk20κ3Wϕ(κ)GD(κ)[J0(ρκ)cos(2β)J2(ρκ)],

where ρ represents the geometrical separation between points in the plane transverse to the direction of propagation, β is the angle between the baseline and the AOA observation axis (see Fig. 1), k = 2π/λ and λ denotes the optical wavelength Wϕ (κ) is the wave-front phase power spectrum, GD(κ) represents the point-spread function of the receiver aperture, and J0(ρκ) and J2(ρκ) denote the zero and second order Bessel functions, respectively.

 

Fig. 1. Schematic layout of the baseline ρ⃗, the transverse wind velocity ν⃗, the observation axis x of the AOA fluctuations, and the angle β between ρ⃗ and x axis.

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Based on Eq. (5), the Taylor frozen turbulence hypothesis allows us to make the association ρ⃗ = ν⃗t, where ν⃗ denotes the transverse wind velocity. In the case the temporal covariance function of AOA can be written as

Cθtβ=πk20κ3Wϕ(κ)GD(κ)[J0(νtκ)cos(2β)J2(νtκ)].

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 phase power spectrum [22, 23] is given by

Wϕ(pl)(κ)=2πk20LΦn(κ)cos2(κ2z2k)dz,

where Φn(κ) is the power spectrum for the refractive-index fluctuations.

For non-Kolmogorov turbulence, Eq. (7) can be written as

Wϕ(pl)κα=2πk20LΦnκαcos2(κ2z2k)dz.

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

Cθ(pl)αtβ=2π20κ3ΦnκαGD(κ)[J0(νtκ)cos(2β)J2(νtκ)]
×0Lcos2(κ2z2k)dzdκ.

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

Wθ(pl)αωβ=8π20κ3ΦnκαGD(κ)[J0(νtκ)cos(2β)J2(νtκ)]
×0Lcos2(κ2z2k)0cos(ωt)dκdzdt.

Using the integral relations [24],

0J0(ax)cos(bx)dx={(a2b2)1/2,0<b<a,0,b>a,

and

0J2n(ax)cos(bx)dx={(1)n(a2b2)1/2T2n(b/a),0<b<a,0,b>a,

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

Tm(z)=cos(mcos1z),

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

Wθ(pl)αωβ=8π20dzω/νκ3ΦnκαGD(κ)cos2(κ2z2k)
×{[(νκ)2ω2]1/2+cos(2β)[(νκ)2ω2]1/2
×[2(ωνκ)21]}.

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

GD(κ)exp(c2D2κ24),

here c = 0.4832.

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

Wθ(pl)αωβ=8π2A(α)C˜n20Ldzω/νκ3αexp(c2D2κ24)
×cos2(κ2z2k){[(νκ)2ω2]1/2+cos(2β)
×[(νκ)2ω2]1/2[2(ωνκ)21]}.

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)1/2D, is satisfied, and considering the integral relation [24]

0(t+a)2μ1(tb)2ν1exp(pt)dt
={0,0<t<b,Γ(2ν)(a+b)μ+ν1pμνep(ab)/2Wμν,μ+ν1/2(bp+ap),t>b,

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

Wθ(pl)αωβ=7.09π2A(α)C˜n2L1+α4k3α4ω0(ωω0)1α2(c2D24)α54
×exp[kc2D28L(ωω0)2]{[1cos(2β)]W3α4,3α4[kc2D24L(ωω0)2]
+2cos(2β)ωω0(kc2D24L)12W1α4,1α4[kc2D24L(ωω0)2]},

where Wμ,ν(z) is Whittaker’s confluent hypergeometric function [26] and ω0 = ν(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)=2πk20LΦn(κ)(zL)2cos2[κ2z(Lz)2kL]dz.

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

Wϕ(sp)κα=2πk20LΦnκα(zL)2cos2[κ2z(Lz)2kL]dz.

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

Cθ(sp)αtβ=2π20κ3ΦnκαGD(κ)[J0(νtκ)cos(2β)J2(νtκ)]
×0Lcos2[κ2z(Lz)2kL](zL)2dzdκ.

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

Wθ(sp)αωβ=2.36π2A(α)C˜n2L1+α4k3α4ω0(ωω0)1α2(c2D24)α54
×exp[kc2D28L(ωω0)2]{[1cos(2β)]W3α4,3α4[kc2D24L(ωω0)2]
+2cos(2β)ωω0(kc2D24L)12W1α4,1α4[kc2D24L(ωω0)2]}.
 

Fig. 2. The temporal power spectrum of AOA fluctuations scaled by the corresponding variance 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.

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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)1/2D. Moreover, the valid range of the power spectrum of the refractive-index fluctuations imposes the constraints l0 ≪ (L/k)1/2L0, L0D, and ω > 2πν/L0 on Eqs. (18) and (22) again.

The temporal power spectra of AOA fluctuations for the plane and spherical waves, normalized 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 for ω>0.1ω0. In addition, for ω<0.1ω0, the slopes of the curves decrease from 0 to -1 with the rising of power law α, while for ω>0.1ω0 they increase from -4 to -3 with the rising of power law 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 turbulence [30] when a is set to 11/3, although Eqs. (18) and (22) is not formally in agreement with the conventional Kolmogorov results.

 

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

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

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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 of the frequency ratio ω/ω0 in Fig. 3 for several values of the receiver aperture D for a particular case. We take α = 11/3 and β = 0. As it is shown in Fig. 3, with the increase of the receiver aperture D, the slopes of the curves decrease in the low-frequency band, while they are almost the same in the high-frequency band. Moreover, it is also shown in Fig. 3 that increasing the receiver aperture can filter higher frequencies, which come directly from the filter function exp [-kc2D2ω2/(8Lω20)] in Eqs. (18) and (22). To research the effect of the observation orientation β on the temporal spectrum, also the normalized temporal spectra of AOA fluctuations for 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 shown in Fig. 4, for ω<0.1ω0, the slopes of the curves decrease with the increase of beta, while for ω>0.1ω0 they increase with the increase of beta. Finally, the comparison of the temporal power spectrum 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 spectrum of AOA fluctuations for a plane wave is the same as that for a spherical wave when the power law α is set to some value.

 

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

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4. Temporal power spectrum of irradiance fluctuations

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

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 irradiance CI (t,L) by

WIωL=40CItLcos(ωt)dt.

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)tL=8π2k20L0κΦn(κ)J0(νtκ)[1cos(κ2(Lz)k)]dκdz.

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

CI(pl)(α,t,L)=8π2k20L0κΦnκαJ0(νtκ)[1cos(κ2(Lz)k)]dκdz.

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

WI(pl)(α,ω,L)=32π2k20Ldz00κΦnκαJ0(νtκ)×[1cos(κ2(Lz)k)]cos(ωt)dt.

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

WI(pl)αωL=16π2k6α2A(α)C˜n2Lα2ω0(ωω0)1αΓ(12)Γ(α12)Γ(α2)Re{1
2F2(1,2α2;32,3α2;i12(ωω0)2)Γ(α2)Γ(1α2)Γ(1+α2)Γ(α12)Γ(3+α2)
×(i12(ωω0)2)α121F1(12;α+32;i12(ωω0)2)}.

where 1F1(a;c;z) is the confluent hypergeometric function of the first kind and 2F2(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)tL=8π2k20L0κΦn(κ)J0(νtκ)[1cos(κ2z(Lz)Lk)]dκdz.

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

CI(sp)(α,t,L)=8π2k20L0κΦnκαJ0(νtκ)[1cos(κ2z(Lz)Lk)]dκdz.

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

WI(sp)(α,ω,L)=32π2k20Ldz00κΦnκαJ0(νtκ)
×[1cos(κ2z(Lz)Lk)]cos(ωt)dt.
 

Fig. 6. The temporal power spectrum of irradiance fluctuations scaled by the corresponding variance 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.

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

WI(sp)αωL=16π2k6α2A(α)C˜n2Lα2ω0(ωω0)1αΓ(12)Γ(α12)Γ(α2)Re{1
2F2(1,2α2;32,3α2;i14(ωω0)2)12Γ(α2)Γ(1α2)Γ(1+α2)Γ(α12)Γ(2+α2)
×(i14(ωω0)2)α121F1(12;α+2α;i14(ωω0)2)}.

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)1/2L0 and ω > 2πν/L0. In addition, since we are mainly concerned with the influence of the variations of spectral power law α on the temporal power spectrum of irradiance fluctuations here, the receiver aperture D is not included in Eqs. (27) and (31). Finally, it is noteworthy that, when the spectral power law α is set to 11/3, Eqs. (27) and (31) match the conventional Kolmogorov results for the plane and spherical waves perfectly.

 

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

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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 frequency ratio ω/ω0 in Fig. 6 for several values of power law α. Observe that the temporal spectrum for a plane wave is similar to that for a spherical wave. The temporal spectrum is essentially constant for ω < ω0, while decaying as ω1-α for ω > ω0. In addition, the temporal spectrum increases with the rising of power law α. The comparison of the temporal power spectrum of a plane wave with that of a spherical wave is shown in Fig. 7 for power law α = 11/3. It is shown that the temporal power spectrum for a spherical wave is smaller than that for a plane wave and that the spherical wave spectrum extends to higher frequencies.

5. Conclusion

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

The results show that the temporal power spectrum of AOA fluctuations for a plane wave is similar to that for a spherical wave. It decays slightly with the frequency for ω<0.1ω0 and decays rapidly for ω>0.1ω0. In addition, for ω<0.1ω0, the slopes of the curves of the normalized temporal spectrum versus ω/ω0 decrease with the rising of power law α, while for ω>0.1ω0 they increase. The temporal spectrum of irradiance fluctuations for a plane wave is also similar to that for a spherical wave. It is essentially constant for ω< ω0, while decaying as 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 a plane wave and that the spherical wave spectrum extends to higher frequencies for some value of alpha. The results will contribute to the investigations of atmospheric turbulence and optical wave 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.

References and links

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References

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  1. 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).
    [CrossRef]
  2. K. Kazaura, K. Omae, T. Suzuki, and M. Matsumoto, "Enhancing performance of next generation FSO communication systems using soft computing-based predictions," Opt. Express 14, 4958-4968 (2006).
    [CrossRef] [PubMed]
  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).
    [CrossRef] [PubMed]
  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).
    [CrossRef]
  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).
    [CrossRef]
  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).
    [CrossRef]
  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 Research 88, 66-77 (2008).
    [CrossRef]
  8. M. S. Belen’kii, E. Cuellar, K. A. Hughes, and V. A. Rye "Experimental study of spatial structure of turbulence at Maui Space Surveillance Site (MSSS)," Proc. SPIE 6304, 63040U-1-12 (2006).
    [CrossRef]
  9. G. K. Batchelor, "Small-scale variation of convected quantities like temperature in turbulent fluid. Part I. General discussion and the case of small conductivity," J. Fluid Mech. 5, 113-133 (1959).
    [CrossRef]
  10. E. Golbraikh and N. S. Kopeika, "Behavior of structure function of refraction coefficients in different trubulent fields," Appl. Opt. 43, 6151-6156 (2004).
    [CrossRef] [PubMed]
  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).
    [CrossRef]
  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).
    [CrossRef]
  15. G. D. Boreman and C. Dainty, "Zernike expansions for non-Kolmogorov turbulence," J. Opt. Soc. Am. A 13, 517-522 (1996).
    [CrossRef]
  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).
    [CrossRef]
  17. M. S. Belen’kii, "Effect of the stratosphere on star image motion," Opt. Lett. 20, 1359-1361 (1995).
    [CrossRef] [PubMed]
  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).
    [CrossRef]
  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).
    [CrossRef]
  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).
    [CrossRef]
  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).
    [CrossRef]
  23. P. Hickson, "Wave-front curvature sensing from a single defocused image," J. Opt. Soc. Am. 11, 1667-1673 (1994).
    [CrossRef]
  24. A. Erd’elyi, 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).
    [PubMed]
  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).
    [CrossRef]
  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).
    [CrossRef]

2009 (1)

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).
[CrossRef]

2008 (1)

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 Research 88, 66-77 (2008).
[CrossRef]

2006 (3)

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).

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).
[CrossRef]

K. Kazaura, K. Omae, T. Suzuki, and M. Matsumoto, "Enhancing performance of next generation FSO communication systems using soft computing-based predictions," Opt. Express 14, 4958-4968 (2006).
[CrossRef] [PubMed]

2004 (1)

2000 (2)

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).
[CrossRef]

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).
[CrossRef]

1997 (1)

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).
[CrossRef]

1996 (3)

S. S. Moiseev and O. G. Chkhetiani, "Helical scaling in turbulence," JETP 83, 192-198 (1996).

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).
[CrossRef]

G. D. Boreman and C. Dainty, "Zernike expansions for non-Kolmogorov turbulence," J. Opt. Soc. Am. A 13, 517-522 (1996).
[CrossRef]

1995 (5)

1994 (2)

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).
[CrossRef]

P. Hickson, "Wave-front curvature sensing from a single defocused image," J. Opt. Soc. Am. 11, 1667-1673 (1994).
[CrossRef]

1991 (1)

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).
[CrossRef]

1971 (1)

1959 (1)

G. K. Batchelor, "Small-scale variation of convected quantities like temperature in turbulent fluid. Part I. General discussion and the case of small conductivity," J. Fluid Mech. 5, 113-133 (1959).
[CrossRef]

Andrews, L. C.

Batchelor, G. K.

G. K. Batchelor, "Small-scale variation of convected quantities like temperature in turbulent fluid. Part I. General discussion and the case of small conductivity," J. Fluid Mech. 5, 113-133 (1959).
[CrossRef]

Be, M. S.

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).
[CrossRef]

Beland, R. R.

R. R. Beland, "Some aspects of propagation through weak isotropic non-Kolmogorov turbulence," Proc. SPIE 2375, 1111-1126 (1995).

Belen’kii, M. S.

Bishop, K. P.

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).
[CrossRef]

Boreman, G. D.

Borgnino, J.

Chkhetiani, O. G.

S. S. Moiseev and O. G. Chkhetiani, "Helical scaling in turbulence," JETP 83, 192-198 (1996).

Chong, Gao

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).

Clifford, S. F.

Conan, R.

Dainty, C.

Du, W.

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).
[CrossRef]

Eaton, F. D.

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).
[CrossRef]

Elperin, T.

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).
[CrossRef]

Golbraikh, E.

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 Research 88, 66-77 (2008).
[CrossRef]

E. Golbraikh and N. S. Kopeika, "Behavior of structure function of refraction coefficients in different trubulent fields," Appl. Opt. 43, 6151-6156 (2004).
[CrossRef] [PubMed]

Gurvich, A. S.

Hammel, S. M.

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).
[CrossRef]

Hickson, P.

P. Hickson, "Wave-front curvature sensing from a single defocused image," J. Opt. Soc. Am. 11, 1667-1673 (1994).
[CrossRef]

Jiang, W.

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).
[CrossRef]

Jiang, Y.

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).
[CrossRef]

Jing, Ma

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).

Kazaura, K.

Keating, D. B.

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).
[CrossRef]

Kleeorin, N.

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).
[CrossRef]

Kopeika, N. S.

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 Research 88, 66-77 (2008).
[CrossRef]

E. Golbraikh and N. S. Kopeika, "Behavior of structure function of refraction coefficients in different trubulent fields," Appl. Opt. 43, 6151-6156 (2004).
[CrossRef] [PubMed]

Kupershmidt, I.

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 Research 88, 66-77 (2008).
[CrossRef]

Kyrazis, D. T.

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).
[CrossRef]

Lachinova, S. L.

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).
[CrossRef]

Ling, N.

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).
[CrossRef]

Liying, Tan

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).

Ma, J.

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).
[CrossRef]

Martin, F.

Matsumoto, M.

Moiseev, S. S.

S. S. Moiseev and O. G. Chkhetiani, "Helical scaling in turbulence," JETP 83, 192-198 (1996).

Omae, K.

Phillips, R. L.

Preble, A. J.

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).
[CrossRef]

Rao, C.

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).
[CrossRef]

Ricklin, J. C.

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).
[CrossRef]

Rogachevskii, I.

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).
[CrossRef]

Roggemann, M. C.

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, "Optical propagation in non-Kolmogorov atmospheric turbulence," Proc. SPIE 2471, 181-196 (1995).
[CrossRef]

Shtemler, Y.

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 Research 88, 66-77 (2008).
[CrossRef]

Stribling, B. E.

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, "Optical propagation in non-Kolmogorov atmospheric turbulence," Proc. SPIE 2471, 181-196 (1995).
[CrossRef]

Suzuki, T.

Tan, L.

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).
[CrossRef]

Virtser, A.

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 Research 88, 66-77 (2008).
[CrossRef]

Welsh, B. M.

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, "Optical propagation in non-Kolmogorov atmospheric turbulence," Proc. SPIE 2471, 181-196 (1995).
[CrossRef]

Winker, D. M.

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).
[CrossRef]

Wissler, J.

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).
[CrossRef]

Xie, W.

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).
[CrossRef]

Yu, P. T.

Yu, S.

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).
[CrossRef]

Ziad, A.

Zilberman, A.

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 Research 88, 66-77 (2008).
[CrossRef]

Appl. Opt. (2)

Atmospheric Research (1)

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 Research 88, 66-77 (2008).
[CrossRef]

High Power Laser and Particle Beams (1)

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).

J. Fluid Mech. (1)

G. K. Batchelor, "Small-scale variation of convected quantities like temperature in turbulent fluid. Part I. General discussion and the case of small conductivity," J. Fluid Mech. 5, 113-133 (1959).
[CrossRef]

J. Mod. Opt. (1)

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).
[CrossRef]

J. Opt. Fiber. Commun. Rep. (1)

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).
[CrossRef]

J. Opt. Soc. Am. (2)

S. F. Clifford, "Temporal-frequency spectra for a spherica wave propagating through atmospheirc turbulence," J. Opt. Soc. Am. 61, 1285-1292 (1971).
[CrossRef]

P. Hickson, "Wave-front curvature sensing from a single defocused image," J. Opt. Soc. Am. 11, 1667-1673 (1994).
[CrossRef]

J. Opt. Soc. Am. A (4)

JETP (1)

S. S. Moiseev and O. G. Chkhetiani, "Helical scaling in turbulence," JETP 83, 192-198 (1996).

Opt. Commun. (1)

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).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. E (1)

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).
[CrossRef]

Proc. SPIE (4)

R. R. Beland, "Some aspects of propagation through weak isotropic non-Kolmogorov turbulence," Proc. SPIE 2375, 1111-1126 (1995).

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, "Optical propagation in non-Kolmogorov atmospheric turbulence," Proc. SPIE 2471, 181-196 (1995).
[CrossRef]

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).
[CrossRef]

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).
[CrossRef]

Other (10)

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).
[CrossRef]

M. S. Belen’kii, E. Cuellar, K. A. Hughes, and V. A. Rye "Experimental study of spatial structure of turbulence at Maui Space Surveillance Site (MSSS)," Proc. SPIE 6304, 63040U-1-12 (2006).
[CrossRef]

L. C. Andrews and R.L. Phillips, Laser Propagation through Random Media (SPIE Optical Engineering Press, Bellingham, 1998).

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).
[CrossRef]

A. Erd’elyi, Tables of Integral Transforms, (McGraw-Hill, New York, 1959).

C. Ho and A. Wheelon, Power Spectrum of Atmospheric Scintillation for the Deep Space Network Goldstone Ka-band Downlink, (Jet Propulsion Laboratory, California, 2004).
[PubMed]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, (Dover, New York, 1965).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic Press, New York, 1978).

W. L. Wolf and G. J. Zissis, The Infrared Handbook (Office of Naval Research, Washington, 1978).

V. I. Tatarski, Wave Propagating in a Turbulent Medium, (McGraw-Hill, New York, 1961).

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Figures (7)

Fig. 1.
Fig. 1.

Schematic layout of the baseline ρ⃗, the transverse wind velocity ν⃗, the observation axis x of the AOA fluctuations, and the angle β between ρ⃗ and x axis.

Fig. 2.
Fig. 2.

The temporal power spectrum of AOA fluctuations scaled by the corresponding variance 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.

Fig. 3.
Fig. 3.

The temporal power spectrum of AOA fluctuations scaled by the corresponding variance as a function of the frequency ratio ω/ω0 for several values of the receiver aperture D, with (a) for a plane wave, (b) for a spherical wave.

Fig. 4.
Fig. 4.

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

Fig. 5.
Fig. 5.

The temporal power spectrum of AOA fluctuations scaled by the corresponding variance as a function of the frequency ratio ω/ω0 for power law α = 11/3. The solid curve represents a spherical wave. The circle denotes a plane wave.

Fig. 6.
Fig. 6.

The temporal power spectrum of irradiance fluctuations scaled by the corresponding variance 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.

Fig. 7.
Fig. 7.

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

Equations (49)

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Φ n κ α = A ( α ) C ˜ n 2 κ α , 2 π / L 0 κ 2 π / l 0 , 3 < α < 4 ,
A ( α ) = 1 4 π 2 Γ ( α 1 ) cos ( απ 2 ) ,
Φ n ( κ ) = 0.033 C n 2 κ 11 / 3 ,
W θ ω β = 4 0 C θ t β cos ( ωt ) dt .
C θ ρ β = π k 2 0 κ 3 W ϕ ( κ ) G D ( κ ) [ J 0 ( ρκ ) cos ( 2 β ) J 2 ( ρκ ) ] ,
C θ t β = π k 2 0 κ 3 W ϕ ( κ ) G D ( κ ) [ J 0 ( νtκ ) cos ( 2 β ) J 2 ( νtκ ) ] .
W ϕ ( pl ) ( κ ) = 2 π k 2 0 L Φ n ( κ ) cos 2 ( κ 2 z 2 k ) dz ,
W ϕ ( pl ) κ α = 2 π k 2 0 L Φ n κ α cos 2 ( κ 2 z 2 k ) dz .
C θ ( pl ) α t β = 2 π 2 0 κ 3 Φ n κ α G D ( κ ) [ J 0 ( νt κ ) cos ( 2 β ) J 2 ( νt κ ) ]
× 0 L cos 2 ( κ 2 z 2 k ) dzdκ .
W θ ( pl ) α ω β = 8 π 2 0 κ 3 Φ n κ α G D ( κ ) [ J 0 ( νtκ ) cos ( 2 β ) J 2 ( νtκ ) ]
× 0 L cos 2 ( κ 2 z 2 k ) 0 cos ( ωt ) dκdzdt .
0 J 0 ( ax ) cos ( bx ) dx = { ( a 2 b 2 ) 1 / 2 , 0 < b < a , 0 , b > a ,
0 J 2 n ( ax ) cos ( bx ) dx = { ( 1 ) n ( a 2 b 2 ) 1 / 2 T 2 n ( b / a ) , 0 < b < a , 0 , b > a ,
T m ( z ) = cos ( m cos 1 z ) ,
W θ ( pl ) α ω β = 8 π 2 0 dz ω / ν κ 3 Φ n κ α G D ( κ ) cos 2 ( κ 2 z 2 k )
× { [ ( νκ ) 2 ω 2 ] 1 / 2 + cos ( 2 β ) [ ( νκ ) 2 ω 2 ] 1 / 2
× [ 2 ( ω νκ ) 2 1 ] } .
G D ( κ ) exp ( c 2 D 2 κ 2 4 ) ,
W θ ( pl ) α ω β = 8 π 2 A ( α ) C ˜ n 2 0 L dz ω / ν κ 3 α exp ( c 2 D 2 κ 2 4 )
× cos 2 ( κ 2 z 2 k ) { [ ( νκ ) 2 ω 2 ] 1 / 2 + cos ( 2 β )
× [ ( νκ ) 2 ω 2 ] 1 / 2 [ 2 ( ω νκ ) 2 1 ] } .
0 ( t + a ) 2 μ 1 ( t b ) 2 ν 1 exp ( pt ) dt
= { 0 , 0 < t < b , Γ ( 2 ν ) ( a + b ) μ + ν 1 p μ ν e p ( a b ) / 2 W μ ν , μ + ν 1 / 2 ( bp + ap ) , t > b ,
W θ ( pl ) α ω β = 7.09 π 2 A ( α ) C ˜ n 2 L 1 + α 4 k 3 α 4 ω 0 ( ω ω 0 ) 1 α 2 ( c 2 D 2 4 ) α 5 4
× exp [ k c 2 D 2 8 L ( ω ω 0 ) 2 ] { [ 1 cos ( 2 β ) ] W 3 α 4 , 3 α 4 [ k c 2 D 2 4 L ( ω ω 0 ) 2 ]
+ 2 cos ( 2 β ) ω ω 0 ( k c 2 D 2 4 L ) 1 2 W 1 α 4 , 1 α 4 [ k c 2 D 2 4 L ( ω ω 0 ) 2 ] } ,
W ϕ ( sp ) ( κ ) = 2 π k 2 0 L Φ n ( κ ) ( z L ) 2 cos 2 [ κ 2 z ( L z ) 2 kL ] dz .
W ϕ ( sp ) κ α = 2 π k 2 0 L Φ n κ α ( z L ) 2 cos 2 [ κ 2 z ( L z ) 2 kL ] dz .
C θ ( sp ) α t β = 2 π 2 0 κ 3 Φ n κ α G D ( κ ) [ J 0 ( νtκ ) cos ( 2 β ) J 2 ( νtκ ) ]
× 0 L cos 2 [ κ 2 z ( L z ) 2 kL ] ( z L ) 2 dzdκ .
W θ ( sp ) α ω β = 2.36 π 2 A ( α ) C ˜ n 2 L 1 + α 4 k 3 α 4 ω 0 ( ω ω 0 ) 1 α 2 ( c 2 D 2 4 ) α 5 4
× exp [ k c 2 D 2 8 L ( ω ω 0 ) 2 ] { [ 1 cos ( 2 β ) ] W 3 α 4 , 3 α 4 [ k c 2 D 2 4 L ( ω ω 0 ) 2 ]
+ 2 cos ( 2 β ) ω ω 0 ( k c 2 D 2 4 L ) 1 2 W 1 α 4 , 1 α 4 [ k c 2 D 2 4 L ( ω ω 0 ) 2 ] } .
W I ω L = 4 0 C I t L cos ( ωt ) dt .
C I ( pl ) t L = 8 π 2 k 2 0 L 0 κ Φ n ( κ ) J 0 ( νtκ ) [ 1 cos ( κ 2 ( L z ) k ) ] dκdz .
C I ( pl ) ( α , t , L ) = 8 π 2 k 2 0 L 0 κ Φ n κ α J 0 ( νtκ ) [ 1 cos ( κ 2 ( L z ) k ) ] dκdz .
W I ( pl ) ( α , ω , L ) = 32 π 2 k 2 0 L dz 0 0 κ Φ n κ α J 0 ( νtκ )
× [ 1 cos ( κ 2 ( L z ) k ) ] cos ( ωt ) dt .
W I ( pl ) α ω L = 16 π 2 k 6 α 2 A ( α ) C ˜ n 2 L α 2 ω 0 ( ω ω 0 ) 1 α Γ ( 1 2 ) Γ ( α 1 2 ) Γ ( α 2 ) Re { 1
2 F 2 ( 1 , 2 α 2 ; 3 2 , 3 α 2 ; i 1 2 ( ω ω 0 ) 2 ) Γ ( α 2 ) Γ ( 1 α 2 ) Γ ( 1 + α 2 ) Γ ( α 1 2 ) Γ ( 3 + α 2 )
× ( i 1 2 ( ω ω 0 ) 2 ) α 1 2 1 F 1 ( 1 2 ; α + 3 2 ; i 1 2 ( ω ω 0 ) 2 ) } .
C I ( sp ) t L = 8 π 2 k 2 0 L 0 κ Φ n ( κ ) J 0 ( νtκ ) [ 1 cos ( κ 2 z ( L z ) Lk ) ] dκdz .
C I ( sp ) ( α , t , L ) = 8 π 2 k 2 0 L 0 κ Φ n κ α J 0 ( νtκ ) [ 1 cos ( κ 2 z ( L z ) Lk ) ] dκdz .
W I ( sp ) ( α , ω , L ) = 32 π 2 k 2 0 L dz 0 0 κ Φ n κ α J 0 ( νtκ )
× [ 1 cos ( κ 2 z ( L z ) Lk ) ] cos ( ωt ) dt .
W I ( sp ) α ω L = 16 π 2 k 6 α 2 A ( α ) C ˜ n 2 L α 2 ω 0 ( ω ω 0 ) 1 α Γ ( 1 2 ) Γ ( α 1 2 ) Γ ( α 2 ) Re { 1
2 F 2 ( 1 , 2 α 2 ; 3 2 , 3 α 2 ; i 1 4 ( ω ω 0 ) 2 ) 1 2 Γ ( α 2 ) Γ ( 1 α 2 ) Γ ( 1 + α 2 ) Γ ( α 1 2 ) Γ ( 2 + α 2 )
× ( i 1 4 ( ω ω 0 ) 2 ) α 1 2 1 F 1 ( 1 2 ; α + 2 α ; i 1 4 ( ω ω 0 ) 2 ) } .

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