Abstract

Kolmogorov turbulence theory based models cannot be directly applied in non-Kolmogorov turbulence case, which has been reported recently by increasing experimental evidence and theoretical investigation. In this study, based on the generalized von Karman spectral model, the theoretical expression of the irradiance scintillation index is derived for Gaussian-beam wave propagating through weak non-Kolmogorov turbulence with horizontal path. In the derivation, the expression is divided into two parts for physical analysis purpose and mathematical analysis convenience. This expression considers the influences of finite turbulence inner and outer scales and has a general spectral power law value in the range 3 to 4 instead of standard power law value of 11/3 (for Kolmogorov turbulence). Numerical simulations are conducted to investigate the influences.

© 2011 OSA

1. Introduction

The performance of a laser radar or laser communication system can be significantly degraded by turbulence-induced scintillation resulting from beam propagating through the atmosphere [14]. Specifically, irradiance scintillation can lead to power losses at the receiver and eventually to fading of the received signal below a prescribed threshold. Over the past decades, several atmospheric turbulence spectral models have been developed and applied in the research of the irradiance scintillation index associated with Gaussian-beam wave propagating through Kolmogorov atmospheric turbulence [510]. Only the modified atmospheric spectral model [11] can feature the high frequency enhancement property (also called “bump” property, which is caused by the turbulence inner scale) in the irradiance scintillation index [12].

For non-Kolmogorov turbulence existing in certain portions of the atmosphere [1318], the general non-Kolmogorov spectral model has been proposed and used to investigate the irradiance scintillation for Gaussian-beam wave propagating through weak non-Kolmogorov turbulence [19,20]. However, this spectral model does not consider the influence of finite turbulence inner and outer scales. To study non-Kolmogorov atmospheric turbulence, some theoretical spectral models were also developed, such as the generalized von Karman spectrum [21] and the generalized Exponential spectrum [22], which consider finite turbulence inner and outer scales and have general spectral power law values instead of the standard power law value of 11/3. The generalized modified atmospheric spectral model [23] for the non-Kolmogorov turbulence can characterize the high frequency enhancement property, but some coefficients in this spectral model should be reevaluated and justified by future experimental data.

In this study, the generalized von Karman spectral model is used to investigate the irradiance scintillation for Gaussian-beam wave propagating through weak non-Kolmogorov turbulence. And then, the impacts of turbulence inner scales, outer scales and spectral power law values on the irradiance scintillation index have been analyzed.

2. Generalized von Karman spectrum

The generalized von Karman spectral model [21] has the following form

Φn(κ,α,l0,L0)=A(α)C^n2(κ2+κ02)α/2exp(κ2κl2)(0κ<,3<α<4).
where α is the spectral power law depending on physical conditions in the given atmospheric layer, and varies from 3 to 4,C^n2 is the generalized refractive-index structure constant (with units of m3α, when α=11/3, with units of m2/3), κ denotes the magnitude of the spatial-frequency vector with units of rad/m and is related to the size of turbulence cells, κl=c(α)/l0,κ0=C0/L0, C0=4π, l0and L0 are the turbulence inner scale and outer scales, respectively. A(α) and c(α)are given by [21]

A(α)=Γ(α1)4π2cos[απ2],c(α)=[Γ(5α2)A(α)23π]1α5.

3. Irradiance scintillation index for Gaussian-beam wave

The irradiance scintillation index for Gaussian-beam wave propagating through weak atmospheric turbulence is given by [24]

σI2(ρ)=8π2k2L010κΦn(κ)exp(ΛLκ2ξ2/k)×{I0(2Λρκξ)cos[Lκ2kξ(1Θ˜ξ)]}dκdξ.
where ρ is the distance from the beam center line in the plane perpendicular to the propagation direction(z axis), L is the wave propagation distance, ξ is the normalized path coordinate and related to z by ξ=1z/L,k=2π/λ,λ is the wavelength, I0(x) is the modified Bessel function of the first kind and zero order, Θ˜ is introduced for the following mathematical analysis convenience, Θ˜=1Θ,Θ and Λ are the output plane (or receiver) beam parameters which are related to the input plane (or transmitter) beam parameters Θ0 and Λ0 [24]
Θ=1+LF=Θ0Θ02+Λ02,Λ=2LkW2=Λ0Θ02+Λ02,
Θ0=1LF0,Λ0=2LkW02.
Here, W0 is the beam radius and F0 is the phase front radius of curvature at the exit aperture of the transmitter. W is the beam radius and F is the phase front radius of curvature at the receiver plane.

For interpretation purpose, σI2(ρ) is expressed as a sum of radial and longitudinal components

σI2(ρ)=σI,r2(ρ)+σI,l2.
where σI,r2(ρ) and σI,l2 represent the radial component and longitudinal component of the irradiance scintillation index, respectively [24]

σI,r2(ρ)=8π2k2L010κΦn(κ)exp(ΛLκ2ξ2/k){I0(2Λρκξ)1}dκdξ,
σI,l2=8π2k2L010κΦn(κ)exp(ΛLκ2ξ2/k){1cos[Lκ2kξ(1Θ˜ξ)]}dκdξ.

Physically, σI,r2(ρ) and σI,l2 describes the off-axis and on-axis contribution to the irradiance scintillation index, respectively. σI,r2(ρ) varies with ρ and becomes zero at the beam centerline (ρ=0). σI,l2 is a constant in the transverse plane at . When Λ=0andΘ=1, σI,r2(ρ)becomes zero (I0(0)1=0) and σI,l2 reduces to the expression of irradiance scintillation index for plane wave. When Λ=Θ=0, σI,r2(ρ)becomes zero and σI,l2 reduces to the expression of irradiance scintillation index for spherical wave.

To consider the influences of finite turbulence inner scale, outer scale and spectral power law values, in the next section, σI2(ρ) is replaced withσI2(ρ,α,l0,L0), and its theoretical expression for Gaussian-beam wave propagating through weak non-Kolmogorov turbulence with horizontal path will be derived.

3.1 The radial component of the irradiance scintillation index

For non-Kolmogorov turbulence, substituting Eq. (1) into Eq. (7), σI,r2(ρ) becomes

σI,r2(ρ,α,l0,L0)=8π2k2L010κΦn(κ,α,l0,L0)exp(ΛLκ2ξ2/k){I0(2Λρκξ)1}dκdξ.

Expanding I0 within a Maclaurin series [25]

I0(x)=J0(ix)=n=0(x/2)2nn!Γ(n+1),

Equation (9) can be expressed as

σI,r2(ρ,α,l0,L0)=8π2k2A(α)C^n2L01{n=1(Λρξ)2nn!Γ(n+1)0κ2n+1exp(κ2/κl2)(κ2+κ02)α/2dκ}dξ.

Integrating with respect to ξ, Eq. (11) becomes

σI,r2(ρ,α,l0,L0)=8π2k2A(α)C^n2Ln=1(Λρ)2nn!Γ(n+1)(2n+1)0κ2n+1exp(κ2/κl2)(κ2+κ02)α/2dκ.

Using the confluent hypergeometric function of the second kind U(a;c;z) and its property [25]

U(a;c;z)=1Γ(a)0eztta1(1+t)ca1dt,
U(a;c;z)Γ(1c)Γ(1+ac)+Γ(c1)Γ(a)z1c,|z|1,

Equation (12) can be expressed as

σI,r2(ρ,α,l0,L0)=4π2k2A(α)C^n2Ln=1(Λρ)2nn!(2n+1)×[κ0α+2+2nΓ(1n+α/2)Γ(α/2)+Γ(nα/2+1)Γ(n+1)(1κl2+ΛLξ2k)1n+α/2].

For analysis purpose, Eq. (15) is divided into two parts

σI,r2(ρ,α,l0,L0)=σI,r12(ρ,α,l0,L0)+σI,r22(ρ,α,l0,L0),
σI,r12(ρ,α,l0,L0)=4π2k2A(α)C^n2Ln=1(Λρ)2nn!(2n+1)[κ0α+2+2nΓ(1n+α/2)Γ(α/2)],
σI,r22(ρ,α,l0,L0)=4π2k2A(α)C^n2Ln=1(Λρ)2nn!(2n+1)[Γ(nα/2+1)Γ(n+1)(1κl2+ΛLξ2k)1n+α/2].

For.., Eq. (17) can be approximated by the simpler expression

σI,r12(ρ,α,l0,L0)=4π2k2A(α)C^n2LΛ2ρ2κ04αΓ(2+α/2)3Γ(α/2),

For mathematical analysis convenience, defining outer scale parameter Q0=Lκ02k, Eq. (19) becomes

σI,r12(ρ,α,l0,L0)=8Γ(2+α/2)3Γ(α/2)π2A(α)C^n2k3α/2Lα/2Q02α/2Λρ2W2.

Defining inner scale parameterQl=Lκl2k, using the gauss hypergeometric function F21(A,B;C;Z) [25]

F21(a,b;c;z)=Γ(c)Γ(b)Γ(cb)01tb1(1t)cb1(1tz)adt,
F21(a,b;c;z)=(1+z)aF21(a,cb;c;z1+z),
and for ρ/W<1, Eq. (18) becomes

σI,r22(ρ,α,l0,L0)=8Γ(2α/2)3π2A(α)C^n2k3α/2Lα/2×Ql2α/2F21(2α2,32;52;ΛQl)Λρ2W2.

Substituting Eqs. (20) and (23) into Eq. (16), σI,r2(ρ,α,l0,L0) can be expressed as

σI,r2(ρ,α,l0,L0)=α3Γ(1α/2)sin(απ/4)σI_pl2(α)Λα/21ρ2W2[Γ(2α2)×(ΛQl)2α/2F21(2α2,32;52;ΛQl)+Γ(2+α/2)Γ(α/2)(ΛQ0)2α/2],
where σI_pl2(α) is the irradiance scintillation index for plane wave propagating through weak non-Kolmogorov turbulence [19]

σI_pl2(α)=8Γ(1α/2)αsin(απ4)π2A(α)C^n2k3α/2Lα/2.

3.2 The longitudinal component of the irradiance scintillation index

For non-Kolmogorov turbulence, substituting Eq. (1) into Eq. (8), σI,l2 becomes

σI,l2(α,l0,L0)=8π2k2L010κΦn(κ,α,l0,L0)×exp(ΛLκ2ξ2/k){1cos[Lκ2kξ(1Θ˜ξ)]}dκdξ.

Usingcos(x)=Re[eix], Eq. (26) becomes

σI,l2(α,l0,L0)=8π2k2L010κ(κ2+κ02)α/2{exp[(ΛLξ2k+1κl2)κ2]Re{exp[(ΛLξ2k+1κl2+iLkξ(1Θ˜ξ))κ2]}}dκdξ.

Using Eqs. (13) and (21), Eq. (27) becomes

σI,l2(α,l0,L0)=4Γ(1α2)π2A(α)C^n2k3α2Lα2Ql1α2{F21(1α2,12;32;ΛQl)Re[01(1+ΛQlξ2+iQlξ(1Θ˜ξ))1+α/2dξ]}.

Following the same procedure as [7], the approximate expression of Eq. (28) can be expressed as

σI,l2(α,l0,L0)=4Γ(1α2)π2A(α)C^n2k3α2Lα2Ql1α2×{F21(1α2,12;32;ΛQl)Re[[1+23ΛQl+iQl(123Θ˜)]α/21α2Ql[23Λ+i(123Θ˜)]]},

ConsideringΘ˜=1Θ, Eq. (29) becomes

σI,l2(α,l0,L0)=4Γ(1α2)π2A(α)C^n2k3α2Lα2Ql1α2{F21(1α2,12;32;ΛQl)2αRe({1+Ql[2Λ/3+i(1+2Θ)/3]}α/21Ql[2Λ/3+i(1+2Θ)/3])},

Here, the part of Re()in Eq. (30) can be expressed as

Re({1+Ql[2Λ/3+i(1+2Θ)/3]}α/21Ql[2Λ/3+i(1+2Θ)/3])=Qlα/21[(1+2Θ)2+(2Λ+3/Ql)]α/43α/21[(1+2Θ)2+4Λ2]1/2sin(α2φ1+φ2)6ΛQl[(1+2Θ)2+4Λ2],
where

φ1=tan1[(1+2Θ)Ql3+2ΛQl],φ2=tan1[2Λ1+2Θ],

Substituting Eq. (31) into Eq. (30), σI,l2(α,l0,L0) becomes

σI,l2(α,l0,L0)=σI_pl2(α)sin(απ/4){[(1+2Θ)2+(2Λ+3/Ql)2]α/43α/21[(1+2Θ)2+4Λ2]1/2sin(α2φ1+φ2)6ΛQlα/2[(1+2Θ)2+4Λ2]α2Ql1α2F21(1α2,12;32;ΛQl)}.

3.3 Irradiance scintillation index for Gaussian-beam wave

For non-Kolmogorov turbulence, substituting Eqs. (24) and (33) into Eq. (6), the expression of σI2(ρ,α,l0,L0) is obtained

σI2(ρ,α,l0,L0)=σI,r2(ρ,α,l0,L0)+σI,l2(α,l0,L0).

Since plane and spherical waves are both characterized by the limiting conditionΛ=0, in this case σI,r2(ρ,α,l0,L0)vanishes, σI2(ρ,α,l0,L0) is given by the longitudinal component

σI2(ρ,α,l0,L0)=σI_pl2(α)sin(απ/4){[(1+2Θ)2+(3/Ql)2]α/43α/21(1+2Θ)sin(α2φ1+φ2)α2Ql1α2}.
Here,Θ=1for plane wave and Θ=0for spherical wave.

4. Numerical results

In this section, simulations are conducted to analyze the influences of l0,L0 and α on the irradiance scintillation index σI2(ρ,α,l0,L0). To avoid the mutual interferences between parameters, in the following simulations, two of the three parameters are fixed and only one parameter’s influence on σI2(ρ,α,l0,L0) is analyzed.

To remove dependence on the structure constantC^n2, the scaled irradiance scintillation index σI2(ρ,α,l0,L0)/σI_pl2(α) as a function of Λ0will be plotted just as the Kolmogorov case [7]. Since the path length L and optical wavelength λ are fixed (L=250mandλ=1.06μm, then L/k=0.0065. In fact, other values can also be chosen), all changes at the transmitter Λ0=2L/kW02 correspond to variations in the transmitter beam radiusW0.

4.1 Effect of inner scale’s variation on the irradiance scintillation index

To analyze turbulence inner scale’s influence on the irradiance scintillation index, α and L0 are assigned to constant values (any α satisfies 3<α<4 can be chosen, in this simulation, α=10/3. For the atmospheric turbulence, usually L0 is in the order of magnitude of meter. L0=1.7m is chosen like [7]). For analysis purpose, different Ql values (introduced immediately following below Eq. (20), it is related to l0 and represents the square ratio of the Fresnel zone L/kwith l0) are chosen instead of various inner scale sizes. Here, Ql=10(l0 is comparable with the Fresnel zone), Ql=100(l0is smaller than the Fresnel zone) and Ql=1000(l0is much smaller than the Fresnel zone and near to zero) are adopted. Other Ql values can also be set in the numerical analysis.

Figure 1 shows σI2(ρ,α,l0,L0)/σI_pl2(α) as a function of Λ0. The lower set of curves represents the on-axis (ρ=0) irradiance scintillation index, while the upper set of curves denotes irradiance scintillation index levels at the diffractive beam edge (ρ/W=1). Also, different types of Gaussian-beam wave are chosen (here collimated beam Θ0=1and convergent beam Θ0<1are chosen). According to Eq. (4), when Λ0=0and Θ0=1, then Λ=0and Θ=1, it corresponds to the plane wave. WhenΛ0, then Λ=0and Θ=0, it denotes the spherical wave no matter what kind of Gaussian-beam wave is chosen.

 

Fig. 1 Scaled irradiance scintillation index as a function of Λ0with different Qlvalues. (a): collimated beam (Θ0=1); (b): convergent beam (Θ0=0.7<1); (c): convergent beam (Θ0=0.1<1).

Download Full Size | PPT Slide | PDF

Figure 1 shows that inner scales effects on the irradiance scintillation index are similar to the Kolmogorov turbulence case (α=11/3) [7] for Λ0 in the range 0 to 100. As l0 increases (Qldecreases), σI2(ρ,α,l0,L0) decreases. This can be explained from the definition of the generalized von Karman spectral modelΦn(κ,α,l0,L0). Whenl0increases, Φn(κ,α,l0,L0)decreases, and that makes σI2(ρ,α,l0,L0) decreases.

This phenomenon can also be explained from the physical point of view: the irradiance fluctuations are contributed mostly by small-scale (<L/k) turbulence cells [10,12]. When l0 increases (Qldecreases), the optical wave meets less small-scale (<L/k) turbulence cells along its propagation path, which makes the irradiance scintillation index decreases.

4.2 Effect of outer scale’s variation on the irradiance scintillation index

In this section, α=10/3, Ql=1000(corresponding to very small l0 value near to zero, and other Qlcan also be chosen). Figure 2 shows σI2(ρ,α,l0,L0)/σI_pl2(α) as a function of Λ0for outer scale L0values of 1m,2mand3m(L0L/kis satisfied).

 

Fig. 2 Scaled irradiance scintillation index as a function of Λ0with different outer scale values. (a): collimated beam (Θ0=1); (b): convergent beam (Θ0=0.7<1); (c): convergent beam (Θ0=0.1<1).

Download Full Size | PPT Slide | PDF

For weak atmospheric turbulence, the irradiance scintillation index for plane wave and spherical wave is not greatly influenced by the outer scale [7,24]. Figure 2 shows that the outer scale has no noticeable effect on the irradiance scintillation near the center of the beam (ρ/W0), similar to the limiting case of plane wave and spherical wave. This can be explained from the physical definition of the irradiance scintillation index for Gaussian beam. When ρ/W0, the radial component of the irradiance scintillation index disappears, and the expression of longitudinal component of the irradiance scintillation (see Eq. (33)) is independent of outer scale L0. However, a finite outer scale of the order of 1-2 m can lead to variation in the irradiance scintillation away from the center of the beam near the diffractive beam edge (ρ/W1), particularly for beams with transmitter diameter ranging from 0.1 to 10 times the size of the Fresnel zone, that is 0.1<Λ0=2LkW02<10. This phenomenon is consistent with the Kolmogorov turbulence case (α=11/3) [7]. It can be explained similar to the physical explanation in section 4.1, here the outer scale’s influence is analyzed instead. When L0 increases, the Gaussian-beam wave meets more small-scale (<L/k) turbulence cells along its propagation path, which makes the irradiance scintillation index increases, especially when the Gaussian-beam width W0is thought to be comparable with the Fresnel zone L/k (in this case, it will produce larger irradiance scintillation index with respect to the case of other Gaussian-beam width [10]).

4.3 Effect of α’s variation on the irradiance scintillation index

In this simulation test, L0 = 1.7m, Ql = 1000. Figure 3 shows that the scaled irradiance scintillation index firstly increases with the increase of α, and then decreases with increasing Λ0. Different values of α lead to more obvious variation in the irradiance scintillation index away from the center of the beam near the diffractive beam edge (ρ/W1) than the case near the center of the beam (ρ/W0).

 

Fig. 3 Scaled irradiance scintillation index as a function of Λ0with different α. (a): collimated beam (Θ0=1); (b): convergent beam (Θ0=0.7<1); (c): convergent beam (Θ0=0.1<1).

Download Full Size | PPT Slide | PDF

5. Conclusions

In this study, theoretical expression of the irradiance scintillation index with finite inner scale, finite outer scale and general spectral power law is derived under the assumption of weak-fluctuation theory for Gaussian-beam wave propagating through non-Kolmogorov atmospheric turbulence with horizontal path.

Simulation results show that variable turbulence outer scale produces obvious effects on the irradiance scintillation index away from the center of the beam near the diffractive beam edge, and produces ignorable effects on the case near the center of the beam, and this conclusion is different from the cases for plane wave and spherical wave [7,24]. Variable turbulence inner scale brings obvious impacts on the final model, as the turbulence inner scale size increases, the irradiance scintillation index decreases. Different power law α produces obvious effects on the irradiance scintillation index especially for the case near the diffractive beam edge. The results in this study will help to better investigate the effects of turbulence on the Gaussian-beam wave propagating through weak non-Kolmogorov atmospheric turbulence with horizontal path.

Acknowledgments

This work is partly supported by the Innovation Funds of BUAA for PhD Students (No.2011115008, No.2011115009), Scholarship Award for Excellent Doctor Student granted by Ministry of Education, and the Fundamental Research Funds for the Central Universities (No.2011115020).

References and links

1. L. C. Andrews and R. L. Phillips, “Impact of scintillation on laser communication systems: recent advances in modeling,” Proc. SPIE 4489, 23–34 (2002). [CrossRef]  

2. X.-W. Qiang, J.-P. Song, J.-W. Feng, and Y. Han, “Irradiance scintillation on laser beam propagation in the near ground turbulent atmosphere,” Proc. SPIE 7382, 73824O (2009). [CrossRef]  

3. W. Cheng, J. W. Haus, and Q. Zhan, “Propagation of vector vortex beams through a turbulent atmosphere,” Opt. Express 17(20), 17829–17836 (2009), http://www.opticsinfobase.org/abstract.cfm?uri=oe-17-20-17829. [CrossRef]   [PubMed]  

4. A. García-Zambrana, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Space-time trellis coding with transmit laser selection for FSO links over strong atmospheric turbulence channels,” Opt. Express 18(6), 5356–5366 (2010), http://www.opticsinfobase.org/abstract.cfm?uri=oe-18-6-5356. [CrossRef]   [PubMed]  

5. R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence (Springer, 1994).

6. 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. A 10(4), 661–672 (1993). [CrossRef]  

7. W. B. Miller, J. C. Ricklin, and L. C. Andrews, “Effects of the refractive index spectral model on the irradiance variance of a Gaussian beam,” J. Opt. Soc. Am. A 11(10), 2719–2726 (1994). [CrossRef]  

8. J. D. Shelton, “Turbulence-induced scintillation on Gaussian-beam waves: theoretical predictions and observations from a laser-illuminated satellite,” J. Opt. Soc. Am. A 12(10), 2172–2181 (1995). [CrossRef]  

9. L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, “Theory of optical scintillation: Gaussian-beam wave model,” Waves Random Complex Media 11, 271–291 (2001). http://dx.doi.org/10.1080/13616670109409785. [CrossRef]  

10. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Optical Engineering Press, 2005), Chap.8.

11. L. C. Andrews, “An analytical model for the refractive index power dpectrum and its spplication to optical scintillations in the atmosphere,” J. Mod. Opt. 39(9), 1849–1853 (1992), http://dx.doi.org/10.1080/09500349214551931. [CrossRef]  

12. V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (trans.for NOAA by Israel Program for Scientific Translations, Jerusalem, 1971).

13. D. T. Kyrazis, J. B. Wissler, D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2120, 43–55 (1994). [CrossRef]  

14. M. S. Belen’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]  

15. 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, 63040U-12 (2006). [CrossRef]  

16. 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,” Atmos. Res. 88(1), 66–77 (2008). [CrossRef]  

17. A. S. Gurvich and M. S. Belen’kii, “Influence of stratospheric turbulence on infrared imaging,” J. Opt. Soc. Am. A 12(11), 2517–2522 (1995). [CrossRef]  

18. M. S. Belen’kii, “Effect of the stratosphere on star image motion,” Opt. Lett. 20(12), 1359–1361 (1995). [CrossRef]   [PubMed]  

19. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for a Gaussian beam propagating through non-Kolmogorov weak turbulence,” IEEE Trans. Antenn. Propag. 57(6), 1783–1788 (2009). [CrossRef]  

20. L. Tan, W. Du, J. Ma, S. Yu, and Q. Han, “Log-amplitude variance for a Gaussian-beam wave propagating through non-Kolmogorov turbulence,” Opt. Express 18(2), 451–462 (2010), http://www.opticsinfobase.org/abstract.cfm?uri=oe-18-2-451. [CrossRef]   [PubMed]  

21. 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, 65510E-12 (2007). [CrossRef]  

22. L. Y. Cui, B. D. Xue, X. G. Cao, J. K. Dong, and J. N. Wang, “Generalized atmospheric turbulence MTF for wave propagating through non-Kolmogorov turbulence,” Opt. Express 18(20), 21269–21283 (2010), http://www.opticsinfobase.org/abstract.cfm?uri=oe-18-20-21269. [CrossRef]   [PubMed]  

23. B. Xue, L. Cui, W. Xue, X. Bai, and F. Zhou, “Generalized modified atmospheric spectral model for optical wave propagating through non-Kolmogorov turbulence,” J. Opt. Soc. Am. A 28(5), 912–916 (2011). [CrossRef]   [PubMed]  

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

25. L. C. Andrews, Special Functions of Mathematics for Engineers, 2nd ed. (SPIE Optical Engineering Press, 1998).

References

  • View by:
  • |
  • |
  • |

  1. L. C. Andrews and R. L. Phillips, “Impact of scintillation on laser communication systems: recent advances in modeling,” Proc. SPIE 4489, 23–34 (2002).
    [CrossRef]
  2. X.-W. Qiang, J.-P. Song, J.-W. Feng, and Y. Han, “Irradiance scintillation on laser beam propagation in the near ground turbulent atmosphere,” Proc. SPIE 7382, 73824O (2009).
    [CrossRef]
  3. W. Cheng, J. W. Haus, and Q. Zhan, “Propagation of vector vortex beams through a turbulent atmosphere,” Opt. Express 17(20), 17829–17836 (2009), http://www.opticsinfobase.org/abstract.cfm?uri=oe-17-20-17829 .
    [CrossRef] [PubMed]
  4. A. García-Zambrana, C. Castillo-Vázquez, and B. Castillo-Vázquez, “Space-time trellis coding with transmit laser selection for FSO links over strong atmospheric turbulence channels,” Opt. Express 18(6), 5356–5366 (2010), http://www.opticsinfobase.org/abstract.cfm?uri=oe-18-6-5356 .
    [CrossRef] [PubMed]
  5. R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence (Springer, 1994).
  6. 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. A 10(4), 661–672 (1993).
    [CrossRef]
  7. W. B. Miller, J. C. Ricklin, and L. C. Andrews, “Effects of the refractive index spectral model on the irradiance variance of a Gaussian beam,” J. Opt. Soc. Am. A 11(10), 2719–2726 (1994).
    [CrossRef]
  8. J. D. Shelton, “Turbulence-induced scintillation on Gaussian-beam waves: theoretical predictions and observations from a laser-illuminated satellite,” J. Opt. Soc. Am. A 12(10), 2172–2181 (1995).
    [CrossRef]
  9. L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, “Theory of optical scintillation: Gaussian-beam wave model,” Waves Random Complex Media 11, 271–291 (2001). http://dx.doi.org/10.1080/13616670109409785 .
    [CrossRef]
  10. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Optical Engineering Press, 2005), Chap.8.
  11. L. C. Andrews, “An analytical model for the refractive index power dpectrum and its spplication to optical scintillations in the atmosphere,” J. Mod. Opt. 39(9), 1849–1853 (1992), http://dx.doi.org/10.1080/09500349214551931 .
    [CrossRef]
  12. V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (trans.for NOAA by Israel Program for Scientific Translations, Jerusalem, 1971).
  13. D. T. Kyrazis, J. B. Wissler, D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2120, 43–55 (1994).
    [CrossRef]
  14. M. S. Belen’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]
  15. 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, 63040U-12 (2006).
    [CrossRef]
  16. 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,” Atmos. Res. 88(1), 66–77 (2008).
    [CrossRef]
  17. A. S. Gurvich and M. S. Belen’kii, “Influence of stratospheric turbulence on infrared imaging,” J. Opt. Soc. Am. A 12(11), 2517–2522 (1995).
    [CrossRef]
  18. M. S. Belen’kii, “Effect of the stratosphere on star image motion,” Opt. Lett. 20(12), 1359–1361 (1995).
    [CrossRef] [PubMed]
  19. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for a Gaussian beam propagating through non-Kolmogorov weak turbulence,” IEEE Trans. Antenn. Propag. 57(6), 1783–1788 (2009).
    [CrossRef]
  20. L. Tan, W. Du, J. Ma, S. Yu, and Q. Han, “Log-amplitude variance for a Gaussian-beam wave propagating through non-Kolmogorov turbulence,” Opt. Express 18(2), 451–462 (2010), http://www.opticsinfobase.org/abstract.cfm?uri=oe-18-2-451 .
    [CrossRef] [PubMed]
  21. 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, 65510E-12 (2007).
    [CrossRef]
  22. L. Y. Cui, B. D. Xue, X. G. Cao, J. K. Dong, and J. N. Wang, “Generalized atmospheric turbulence MTF for wave propagating through non-Kolmogorov turbulence,” Opt. Express 18(20), 21269–21283 (2010), http://www.opticsinfobase.org/abstract.cfm?uri=oe-18-20-21269 .
    [CrossRef] [PubMed]
  23. B. Xue, L. Cui, W. Xue, X. Bai, and F. Zhou, “Generalized modified atmospheric spectral model for optical wave propagating through non-Kolmogorov turbulence,” J. Opt. Soc. Am. A 28(5), 912–916 (2011).
    [CrossRef] [PubMed]
  24. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Optical Engineering Press, 1998).
  25. L. C. Andrews, Special Functions of Mathematics for Engineers, 2nd ed. (SPIE Optical Engineering Press, 1998).

2011 (1)

2010 (3)

2009 (3)

W. Cheng, J. W. Haus, and Q. Zhan, “Propagation of vector vortex beams through a turbulent atmosphere,” Opt. Express 17(20), 17829–17836 (2009), http://www.opticsinfobase.org/abstract.cfm?uri=oe-17-20-17829 .
[CrossRef] [PubMed]

X.-W. Qiang, J.-P. Song, J.-W. Feng, and Y. Han, “Irradiance scintillation on laser beam propagation in the near ground turbulent atmosphere,” Proc. SPIE 7382, 73824O (2009).
[CrossRef]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for a Gaussian beam propagating through non-Kolmogorov weak turbulence,” IEEE Trans. Antenn. Propag. 57(6), 1783–1788 (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,” Atmos. Res. 88(1), 66–77 (2008).
[CrossRef]

2007 (1)

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, 65510E-12 (2007).
[CrossRef]

2006 (1)

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, 63040U-12 (2006).
[CrossRef]

2002 (1)

L. C. Andrews and R. L. Phillips, “Impact of scintillation on laser communication systems: recent advances in modeling,” Proc. SPIE 4489, 23–34 (2002).
[CrossRef]

2001 (1)

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, “Theory of optical scintillation: Gaussian-beam wave model,” Waves Random Complex Media 11, 271–291 (2001). http://dx.doi.org/10.1080/13616670109409785 .
[CrossRef]

1997 (1)

M. S. Belen’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]

1995 (3)

1994 (2)

D. T. Kyrazis, J. B. Wissler, D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2120, 43–55 (1994).
[CrossRef]

W. B. Miller, J. C. Ricklin, and L. C. Andrews, “Effects of the refractive index spectral model on the irradiance variance of a Gaussian beam,” J. Opt. Soc. Am. A 11(10), 2719–2726 (1994).
[CrossRef]

1993 (1)

1992 (1)

L. C. Andrews, “An analytical model for the refractive index power dpectrum and its spplication to optical scintillations in the atmosphere,” J. Mod. Opt. 39(9), 1849–1853 (1992), http://dx.doi.org/10.1080/09500349214551931 .
[CrossRef]

Al-Habash, M. A.

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, “Theory of optical scintillation: Gaussian-beam wave model,” Waves Random Complex Media 11, 271–291 (2001). http://dx.doi.org/10.1080/13616670109409785 .
[CrossRef]

Andrews, L. C.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for a Gaussian beam propagating through non-Kolmogorov weak turbulence,” IEEE Trans. Antenn. Propag. 57(6), 1783–1788 (2009).
[CrossRef]

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, 65510E-12 (2007).
[CrossRef]

L. C. Andrews and R. L. Phillips, “Impact of scintillation on laser communication systems: recent advances in modeling,” Proc. SPIE 4489, 23–34 (2002).
[CrossRef]

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, “Theory of optical scintillation: Gaussian-beam wave model,” Waves Random Complex Media 11, 271–291 (2001). http://dx.doi.org/10.1080/13616670109409785 .
[CrossRef]

W. B. Miller, J. C. Ricklin, and L. C. Andrews, “Effects of the refractive index spectral model on the irradiance variance of a Gaussian beam,” J. Opt. Soc. Am. A 11(10), 2719–2726 (1994).
[CrossRef]

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. A 10(4), 661–672 (1993).
[CrossRef]

L. C. Andrews, “An analytical model for the refractive index power dpectrum and its spplication to optical scintillations in the atmosphere,” J. Mod. Opt. 39(9), 1849–1853 (1992), http://dx.doi.org/10.1080/09500349214551931 .
[CrossRef]

Bai, X.

Belen’kii, M. S.

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, 63040U-12 (2006).
[CrossRef]

M. S. Belen’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]

A. S. Gurvich and M. S. Belen’kii, “Influence of stratospheric turbulence on infrared imaging,” J. Opt. Soc. Am. A 12(11), 2517–2522 (1995).
[CrossRef]

M. S. Belen’kii, “Effect of the stratosphere on star image motion,” Opt. Lett. 20(12), 1359–1361 (1995).
[CrossRef] [PubMed]

Bishop, K. P.

D. T. Kyrazis, J. B. Wissler, D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2120, 43–55 (1994).
[CrossRef]

Brown, J. M.

M. S. Belen’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]

Cao, X. G.

Castillo-Vázquez, B.

Castillo-Vázquez, C.

Cheng, W.

Cuellar, E.

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, 63040U-12 (2006).
[CrossRef]

Cui, L.

Cui, L. Y.

Dong, J. K.

Du, W.

Feng, J.-W.

X.-W. Qiang, J.-P. Song, J.-W. Feng, and Y. Han, “Irradiance scintillation on laser beam propagation in the near ground turbulent atmosphere,” Proc. SPIE 7382, 73824O (2009).
[CrossRef]

Ferrero, V.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for a Gaussian beam propagating through non-Kolmogorov weak turbulence,” IEEE Trans. Antenn. Propag. 57(6), 1783–1788 (2009).
[CrossRef]

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, 65510E-12 (2007).
[CrossRef]

Fugate, R. Q.

M. S. Belen’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]

García-Zambrana, A.

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,” Atmos. Res. 88(1), 66–77 (2008).
[CrossRef]

Gurvich, A. S.

Han, Q.

Han, Y.

X.-W. Qiang, J.-P. Song, J.-W. Feng, and Y. Han, “Irradiance scintillation on laser beam propagation in the near ground turbulent atmosphere,” Proc. SPIE 7382, 73824O (2009).
[CrossRef]

Haus, J. W.

Hopen, C. Y.

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, “Theory of optical scintillation: Gaussian-beam wave model,” Waves Random Complex Media 11, 271–291 (2001). http://dx.doi.org/10.1080/13616670109409785 .
[CrossRef]

Hughes, K. A.

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, 63040U-12 (2006).
[CrossRef]

Karis, S. J.

M. S. Belen’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]

Keating, D. B.

D. T. Kyrazis, J. B. Wissler, D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2120, 43–55 (1994).
[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,” Atmos. Res. 88(1), 66–77 (2008).
[CrossRef]

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,” Atmos. Res. 88(1), 66–77 (2008).
[CrossRef]

Kyrazis, D. T.

D. T. Kyrazis, J. B. Wissler, D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2120, 43–55 (1994).
[CrossRef]

Ma, J.

Miller, W. B.

Phillips, R. L.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for a Gaussian beam propagating through non-Kolmogorov weak turbulence,” IEEE Trans. Antenn. Propag. 57(6), 1783–1788 (2009).
[CrossRef]

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, 65510E-12 (2007).
[CrossRef]

L. C. Andrews and R. L. Phillips, “Impact of scintillation on laser communication systems: recent advances in modeling,” Proc. SPIE 4489, 23–34 (2002).
[CrossRef]

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, “Theory of optical scintillation: Gaussian-beam wave model,” Waves Random Complex Media 11, 271–291 (2001). http://dx.doi.org/10.1080/13616670109409785 .
[CrossRef]

Preble, A. J.

D. T. Kyrazis, J. B. Wissler, D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2120, 43–55 (1994).
[CrossRef]

Qiang, X.-W.

X.-W. Qiang, J.-P. Song, J.-W. Feng, and Y. Han, “Irradiance scintillation on laser beam propagation in the near ground turbulent atmosphere,” Proc. SPIE 7382, 73824O (2009).
[CrossRef]

Ricklin, J. C.

Rye, V. A.

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, 63040U-12 (2006).
[CrossRef]

Shelton, J. D.

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,” Atmos. Res. 88(1), 66–77 (2008).
[CrossRef]

Song, J.-P.

X.-W. Qiang, J.-P. Song, J.-W. Feng, and Y. Han, “Irradiance scintillation on laser beam propagation in the near ground turbulent atmosphere,” Proc. SPIE 7382, 73824O (2009).
[CrossRef]

Tan, L.

Toselli, I.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for a Gaussian beam propagating through non-Kolmogorov weak turbulence,” IEEE Trans. Antenn. Propag. 57(6), 1783–1788 (2009).
[CrossRef]

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, 65510E-12 (2007).
[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,” Atmos. Res. 88(1), 66–77 (2008).
[CrossRef]

Wang, J. N.

Wissler, J. B.

D. T. Kyrazis, J. B. Wissler, D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2120, 43–55 (1994).
[CrossRef]

Xue, B.

Xue, B. D.

Xue, W.

Yu, S.

Zhan, Q.

Zhou, F.

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,” Atmos. Res. 88(1), 66–77 (2008).
[CrossRef]

Atmos. Res. (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,” Atmos. Res. 88(1), 66–77 (2008).
[CrossRef]

IEEE Trans. Antenn. Propag. (1)

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for a Gaussian beam propagating through non-Kolmogorov weak turbulence,” IEEE Trans. Antenn. Propag. 57(6), 1783–1788 (2009).
[CrossRef]

J. Mod. Opt. (1)

L. C. Andrews, “An analytical model for the refractive index power dpectrum and its spplication to optical scintillations in the atmosphere,” J. Mod. Opt. 39(9), 1849–1853 (1992), http://dx.doi.org/10.1080/09500349214551931 .
[CrossRef]

J. Opt. Soc. Am. A (5)

Opt. Express (4)

Opt. Lett. (1)

Proc. SPIE (6)

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, 65510E-12 (2007).
[CrossRef]

D. T. Kyrazis, J. B. Wissler, D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2120, 43–55 (1994).
[CrossRef]

M. S. Belen’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]

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, 63040U-12 (2006).
[CrossRef]

L. C. Andrews and R. L. Phillips, “Impact of scintillation on laser communication systems: recent advances in modeling,” Proc. SPIE 4489, 23–34 (2002).
[CrossRef]

X.-W. Qiang, J.-P. Song, J.-W. Feng, and Y. Han, “Irradiance scintillation on laser beam propagation in the near ground turbulent atmosphere,” Proc. SPIE 7382, 73824O (2009).
[CrossRef]

Waves Random Complex Media (1)

L. C. Andrews, M. A. Al-Habash, C. Y. Hopen, and R. L. Phillips, “Theory of optical scintillation: Gaussian-beam wave model,” Waves Random Complex Media 11, 271–291 (2001). http://dx.doi.org/10.1080/13616670109409785 .
[CrossRef]

Other (5)

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Optical Engineering Press, 2005), Chap.8.

R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence (Springer, 1994).

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (trans.for NOAA by Israel Program for Scientific Translations, Jerusalem, 1971).

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

L. C. Andrews, Special Functions of Mathematics for Engineers, 2nd ed. (SPIE Optical Engineering Press, 1998).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1
Fig. 1

Scaled irradiance scintillation index as a function of Λ 0 with different Q l values. (a): collimated beam ( Θ 0 = 1 ); (b): convergent beam ( Θ 0 = 0.7 < 1 ); (c): convergent beam ( Θ 0 = 0.1 < 1 ).

Fig. 2
Fig. 2

Scaled irradiance scintillation index as a function of Λ 0 with different outer scale values. (a): collimated beam ( Θ 0 = 1 ); (b): convergent beam ( Θ 0 = 0.7 < 1 ); (c): convergent beam ( Θ 0 = 0.1 < 1 ).

Fig. 3
Fig. 3

Scaled irradiance scintillation index as a function of Λ 0 with different α. (a): collimated beam ( Θ 0 = 1 ); (b): convergent beam ( Θ 0 = 0.7 < 1 ); (c): convergent beam ( Θ 0 = 0.1 < 1 ).

Equations (35)

Equations on this page are rendered with MathJax. Learn more.

Φ n ( κ , α , l 0 , L 0 ) = A ( α ) C ^ n 2 ( κ 2 + κ 0 2 ) α / 2 exp ( κ 2 κ l 2 ) ( 0 κ < , 3 < α < 4 ) .
A ( α ) = Γ ( α 1 ) 4 π 2 cos [ α π 2 ] , c ( α ) = [ Γ ( 5 α 2 ) A ( α ) 2 3 π ] 1 α 5 .
σ I 2 ( ρ ) = 8 π 2 k 2 L 0 1 0 κ Φ n ( κ ) exp ( Λ L κ 2 ξ 2 / k ) × { I 0 ( 2 Λ ρ κ ξ ) cos [ L κ 2 k ξ ( 1 Θ ˜ ξ ) ] } d κ d ξ .
Θ = 1 + L F = Θ 0 Θ 0 2 + Λ 0 2 , Λ = 2 L k W 2 = Λ 0 Θ 0 2 + Λ 0 2 ,
Θ 0 = 1 L F 0 , Λ 0 = 2 L k W 0 2 .
σ I 2 ( ρ ) = σ I , r 2 ( ρ ) + σ I , l 2 .
σ I , r 2 ( ρ ) = 8 π 2 k 2 L 0 1 0 κ Φ n ( κ ) exp ( Λ L κ 2 ξ 2 / k ) { I 0 ( 2 Λ ρ κ ξ ) 1 } d κ d ξ ,
σ I , l 2 = 8 π 2 k 2 L 0 1 0 κ Φ n ( κ ) exp ( Λ L κ 2 ξ 2 / k ) { 1 cos [ L κ 2 k ξ ( 1 Θ ˜ ξ ) ] } d κ d ξ .
σ I , r 2 ( ρ , α , l 0 , L 0 ) = 8 π 2 k 2 L 0 1 0 κ Φ n ( κ , α , l 0 , L 0 ) exp ( Λ L κ 2 ξ 2 / k ) { I 0 ( 2 Λ ρ κ ξ ) 1 } d κ d ξ .
I 0 ( x ) = J 0 ( i x ) = n = 0 ( x / 2 ) 2 n n ! Γ ( n + 1 ) ,
σ I , r 2 ( ρ , α , l 0 , L 0 ) = 8 π 2 k 2 A ( α ) C ^ n 2 L 0 1 { n = 1 ( Λ ρ ξ ) 2 n n ! Γ ( n + 1 ) 0 κ 2 n + 1 exp ( κ 2 / κ l 2 ) ( κ 2 + κ 0 2 ) α / 2 d κ } d ξ .
σ I , r 2 ( ρ , α , l 0 , L 0 ) = 8 π 2 k 2 A ( α ) C ^ n 2 L n = 1 ( Λ ρ ) 2 n n ! Γ ( n + 1 ) ( 2 n + 1 ) 0 κ 2 n + 1 exp ( κ 2 / κ l 2 ) ( κ 2 + κ 0 2 ) α / 2 d κ .
U ( a ; c ; z ) = 1 Γ ( a ) 0 e z t t a 1 ( 1 + t ) c a 1 d t ,
U ( a ; c ; z ) Γ ( 1 c ) Γ ( 1 + a c ) + Γ ( c 1 ) Γ ( a ) z 1 c , | z | 1 ,
σ I , r 2 ( ρ , α , l 0 , L 0 ) = 4 π 2 k 2 A ( α ) C ^ n 2 L n = 1 ( Λ ρ ) 2 n n ! ( 2 n + 1 ) × [ κ 0 α + 2 + 2 n Γ ( 1 n + α / 2 ) Γ ( α / 2 ) + Γ ( n α / 2 + 1 ) Γ ( n + 1 ) ( 1 κ l 2 + Λ L ξ 2 k ) 1 n + α / 2 ] .
σ I , r 2 ( ρ , α , l 0 , L 0 ) = σ I , r 1 2 ( ρ , α , l 0 , L 0 ) + σ I , r 2 2 ( ρ , α , l 0 , L 0 ) ,
σ I , r 1 2 ( ρ , α , l 0 , L 0 ) = 4 π 2 k 2 A ( α ) C ^ n 2 L n = 1 ( Λ ρ ) 2 n n ! ( 2 n + 1 ) [ κ 0 α + 2 + 2 n Γ ( 1 n + α / 2 ) Γ ( α / 2 ) ] ,
σ I , r 2 2 ( ρ , α , l 0 , L 0 ) = 4 π 2 k 2 A ( α ) C ^ n 2 L n = 1 ( Λ ρ ) 2 n n ! ( 2 n + 1 ) [ Γ ( n α / 2 + 1 ) Γ ( n + 1 ) ( 1 κ l 2 + Λ L ξ 2 k ) 1 n + α / 2 ] .
σ I , r 1 2 ( ρ , α , l 0 , L 0 ) = 4 π 2 k 2 A ( α ) C ^ n 2 L Λ 2 ρ 2 κ 0 4 α Γ ( 2 + α / 2 ) 3 Γ ( α / 2 ) ,
σ I , r 1 2 ( ρ , α , l 0 , L 0 ) = 8 Γ ( 2 + α / 2 ) 3 Γ ( α / 2 ) π 2 A ( α ) C ^ n 2 k 3 α / 2 L α / 2 Q 0 2 α / 2 Λ ρ 2 W 2 .
F 2 1 ( a , b ; c ; z ) = Γ ( c ) Γ ( b ) Γ ( c b ) 0 1 t b 1 ( 1 t ) c b 1 ( 1 t z ) a d t ,
F 2 1 ( a , b ; c ; z ) = ( 1 + z ) a F 2 1 ( a , c b ; c ; z 1 + z ) ,
σ I , r 2 2 ( ρ , α , l 0 , L 0 ) = 8 Γ ( 2 α / 2 ) 3 π 2 A ( α ) C ^ n 2 k 3 α / 2 L α / 2 × Q l 2 α / 2 F 2 1 ( 2 α 2 , 3 2 ; 5 2 ; Λ Q l ) Λ ρ 2 W 2 .
σ I , r 2 ( ρ , α , l 0 , L 0 ) = α 3 Γ ( 1 α / 2 ) sin ( α π / 4 ) σ I _ p l 2 ( α ) Λ α / 2 1 ρ 2 W 2 [ Γ ( 2 α 2 ) × ( Λ Q l ) 2 α / 2 F 2 1 ( 2 α 2 , 3 2 ; 5 2 ; Λ Q l ) + Γ ( 2 + α / 2 ) Γ ( α / 2 ) ( Λ Q 0 ) 2 α / 2 ] ,
σ I _ p l 2 ( α ) = 8 Γ ( 1 α / 2 ) α sin ( α π 4 ) π 2 A ( α ) C ^ n 2 k 3 α / 2 L α / 2 .
σ I , l 2 ( α , l 0 , L 0 ) = 8 π 2 k 2 L 0 1 0 κ Φ n ( κ , α , l 0 , L 0 ) × exp ( Λ L κ 2 ξ 2 / k ) { 1 cos [ L κ 2 k ξ ( 1 Θ ˜ ξ ) ] } d κ d ξ .
σ I , l 2 ( α , l 0 , L 0 ) = 8 π 2 k 2 L 0 1 0 κ ( κ 2 + κ 0 2 ) α / 2 { exp [ ( Λ L ξ 2 k + 1 κ l 2 ) κ 2 ] Re { exp [ ( Λ L ξ 2 k + 1 κ l 2 + i L k ξ ( 1 Θ ˜ ξ ) ) κ 2 ] } } d κ d ξ .
σ I , l 2 ( α , l 0 , L 0 ) = 4 Γ ( 1 α 2 ) π 2 A ( α ) C ^ n 2 k 3 α 2 L α 2 Q l 1 α 2 { F 2 1 ( 1 α 2 , 1 2 ; 3 2 ; Λ Q l ) Re [ 0 1 ( 1 + Λ Q l ξ 2 + i Q l ξ ( 1 Θ ˜ ξ ) ) 1 + α / 2 d ξ ] } .
σ I , l 2 ( α , l 0 , L 0 ) = 4 Γ ( 1 α 2 ) π 2 A ( α ) C ^ n 2 k 3 α 2 L α 2 Q l 1 α 2 × { F 2 1 ( 1 α 2 , 1 2 ; 3 2 ; Λ Q l ) Re [ [ 1 + 2 3 Λ Q l + i Q l ( 1 2 3 Θ ˜ ) ] α / 2 1 α 2 Q l [ 2 3 Λ + i ( 1 2 3 Θ ˜ ) ] ] } ,
σ I , l 2 ( α , l 0 , L 0 ) = 4 Γ ( 1 α 2 ) π 2 A ( α ) C ^ n 2 k 3 α 2 L α 2 Q l 1 α 2 { F 2 1 ( 1 α 2 , 1 2 ; 3 2 ; Λ Q l ) 2 α Re ( { 1 + Q l [ 2 Λ / 3 + i ( 1 + 2 Θ ) / 3 ] } α / 2 1 Q l [ 2 Λ / 3 + i ( 1 + 2 Θ ) / 3 ] ) } ,
Re ( { 1 + Q l [ 2 Λ / 3 + i ( 1 + 2 Θ ) / 3 ] } α / 2 1 Q l [ 2 Λ / 3 + i ( 1 + 2 Θ ) / 3 ] ) = Q l α / 2 1 [ ( 1 + 2 Θ ) 2 + ( 2 Λ + 3 / Q l ) ] α / 4 3 α / 2 1 [ ( 1 + 2 Θ ) 2 + 4 Λ 2 ] 1 / 2 sin ( α 2 φ 1 + φ 2 ) 6 Λ Q l [ ( 1 + 2 Θ ) 2 + 4 Λ 2 ] ,
φ 1 = tan 1 [ ( 1 + 2 Θ ) Q l 3 + 2 Λ Q l ] , φ 2 = tan 1 [ 2 Λ 1 + 2 Θ ] ,
σ I , l 2 ( α , l 0 , L 0 ) = σ I _ p l 2 ( α ) sin ( α π / 4 ) { [ ( 1 + 2 Θ ) 2 + ( 2 Λ + 3 / Q l ) 2 ] α / 4 3 α / 2 1 [ ( 1 + 2 Θ ) 2 + 4 Λ 2 ] 1 / 2 sin ( α 2 φ 1 + φ 2 ) 6 Λ Q l α / 2 [ ( 1 + 2 Θ ) 2 + 4 Λ 2 ] α 2 Q l 1 α 2 F 2 1 ( 1 α 2 , 1 2 ; 3 2 ; Λ Q l ) } .
σ I 2 ( ρ , α , l 0 , L 0 ) = σ I , r 2 ( ρ , α , l 0 , L 0 ) + σ I , l 2 ( α , l 0 , L 0 ) .
σ I 2 ( ρ , α , l 0 , L 0 ) = σ I _ p l 2 ( α ) sin ( α π / 4 ) { [ ( 1 + 2 Θ ) 2 + ( 3 / Q l ) 2 ] α / 4 3 α / 2 1 ( 1 + 2 Θ ) sin ( α 2 φ 1 + φ 2 ) α 2 Q l 1 α 2 } .

Metrics