Abstract

The effects of anisotropic, non-Kolmogorov turbulence on propagating stochastic electromagnetic beam-like fields are discussed for the first time. The atmosphere of interest can be found above the boundary layer, at high (more than 2 km above the ground) altitudes where the energy distribution among the turbulent eddies might not satisfy the classic assumption represented by the famous 11/3 Kolmogorov’s power law, and the anisotropy in the direction orthogonal to the Earth surface is possibly present. Our analysis focuses on the classic electromagnetic Gaussian Schell-model beams but can either be readily reduced to scalar and/or coherent beams or generalized to other beam classes. In particular, we explore the effects of the anisotropic parameter on the spectral density, the spectral degree of coherence and on the spectral degree of polarization of the beam.

© 2014 Optical Society of America

1. Introduction

The atmosphere can be divided into two parts: the atmospheric boundary layer (up to 2 km in altitude), where heating of the surface leads to convective instability, and the free atmosphere (above the atmospheric boundary layer) where the effect of the Earth’s surface friction on the air motion is negligible. Inside the atmospheric boundary layer optical turbulence can be considered homogeneous and isotropic therefore the Kolmogorov model for power spectrum of the fluctuating refractive index is generally valid within the inertial sub-range. However, in the free atmosphere, in particular inside the stably layered stratosphere, optical turbulence can be anisotropic (mostly at large scales) and the Kolmogorov power spectrum does not properly describe the real turbulence behavior. Although some evidence of anisotropy was measured more than forty years ago by Consortini [1] near the ground (about 1 m above), the evidence of anisotropy of optical turbulence is more likely in zones of the atmosphere with inversion layers, i.e. in the presence of the positive temperature gradients, where the vertical component of turbulence is strongly reduced. Dalaudeier et al. [2] showed experimentally the presence in the atmosphere, at least up to 25 km, of very strong positive temperature gradients within very thin layers, termed “sheets”. Such sheets are anisotropic having vertical extension up to 10 m and horizontal extensions larger than 100 m. It is well known that temperature fluctuations yield to optical turbulence, therefore if the temperature field is layered or anisotropic the turbulence field should also assume the same behavior. Grechko et al. [3] reported a strong anisotropy of temperature and density in the middle atmosphere from experimental observations of star scintillation. Biferale et al. [4] used probes with two different geometries (horizontally and vertically) to detect information about anisotropic turbulence in the boundary layer and concluded that the atmospheric boundary layer exhibits three-dimensional statistical turbulence intermingled with flow patterns whose statistics have a quasi-two-dimensional nature. Belenkii et al. [5,6] experimentally observed anisotropy of the wavefront tilt’s statistics. They observed that the horizontal outer scale is bigger than the vertical one, on-axis tilt variances are unequal and the horizontal tilt variance is consistently greater than the vertical one. Also, the evidence of anisotropy in the stratosphere based on the balloon-borne measurements has been reported in [7], where the authors employed a power spectrum model with two components: anisotropic and isotropic. Results showed a major contribution to scintillation of the anisotropic component relative to the isotropic one. Experimental measurements [8,9] have implied that the outer scale of turbulence in the horizontal direction can be many times larger than that in the vertical direction. The horizontal size of these eddies is typically tens of meters across or, in some cases, kilometers across [10]. In vertical direction the size of the outer scale cells is usually confined to a few meters. In addition to anisotropy, turbulence does not always follow the Kolmogorov power spectrum density model but sometimes it follows different power laws [11–13]. Kyrazis et al. measured non-Kolmogorov turbulence in the upper troposphere and lower stratosphere [14–17].

On the other hand, the interaction of deterministic and random optical beams [18–20] with turbulent atmosphere has been the active research area during several decades (see [21–23] and references wherein). In all these studies the atmospheric fluctuations in the refractive index have been assumed to be homogeneous and isotropic which allowed the treatment of beam propagation with the help of the power spectrum depending on the scalar wave number. In particular the details of beam evolution in non-Kolmogorov (isotropic) turbulence were discussed in [24–30]. So far the modeling of anisotropic turbulence [31] (see also [32]) has only resulted in analysis of scalar deterministic beams propagating in it [10], [12] and [33]. The degree of anisotropy has been shown to play an important part in predicting the evolution of such beams.

The objective of this paper is to extend the analysis of anisotropic turbulence from scalar deterministic to electromagnetic stochastic beams and to explore the influence of the anisotropy on all the major second-order statistics along the propagation path. We will restrict our considerations only to the case when the beam travels along the anisotropy axis, i.e. only along up or down link. Therefore our results will be of interest for optimization of the earth-satellite communication and imaging links as well as in astronomical studies. In such cases light must travel for large (tens of kilometers) distances through the upper atmospheric layers where the effects of anisotropy become substantial. Since the sunlight and beams used in communications are in general partially coherent and partially polarized our results are of importance, being able to cover a large variety of possible coherence and polarization states. All the previous studies that included atmospheric anisotropy only treated coherent and fully polarized radiation.

Recently a variety of stochastic sources and beams have been introduced [34–44] all differing by the shape of either the intensity distribution or the degree of coherence. Most of these source and beam models have been derived on the basis of the sufficient condition developed in [45–47]. In this paper however we will restrict ourselves to the classic electromagnetic Gaussian Schell-model beam [18, 23] in order to focus on the effects stemming from atmospheric anisotropy rather than on different source properties. After establishing the propagation laws for the cross-spectral density matrix of the beam we investigate in details the changes in its derivatives: the spectral density, the degree of coherence and the polarization features. The examples pertaining to the limiting case of a coherent beam are presented first followed by those for general electromagnetic random beam.

2. Anisotropic power spectrum with inner and outer scale

For our presentation we will employ the anisotropic non-Kolmogorov power spectrum reported in [33] also including the inner and outer scale effects by using a generalized von Karman model [48]. We assume that the anisotropy exists only along the direction of propagation, say z, of the beam and accounted via an effective anisotropic factor ζas discussed in [32,33]. Hence the power spectrum of refractive-index fluctuations is given by the expression:

Φn(κ,α)=A(α)C˜n2ζ2(ζ2κxy2+κz2+κ02)α2exp(ζ2κxy2+κz2κm2),κ>0,3<α<5,
where κ=ζ2(κx2+κy2)+κz2=ζ2κxy2+κz2, α is the power law, κ0=2π/L0, κm=C(α)/l0, l0 is the inner scale, L0 is the outer scale, C˜n2=βCn2 is a generalized structure parameter with unit [m3α], β is a dimensional constant with unit [m11/3α], ζ is the effective anisotropic factor, C(α) is given by [48]
C(α)={πA(α)Γ(32α2)(3α3)}1α5,3<α<5,
and A(α) is defined by [24]
A(α)=Γ(α1)4π2cos(απ2),3<α<5,
where the symbol Γ(x)stands for Gamma function. For Kolmogorov power law, α=11/3, the generalized structure parameter reduces to the structure parameter Cn2 with unit [m2/3]. Power spectrum (1) is basically a generalized von Karman power spectrum with an anisotropic factor ζ which introduces the turbulence rescaling due to anisotropy along the z-direction. In particular, on setting the inner scale to zero and the outer scale to infinity we deduce that such a rescaling appears as a factor ζ2α.

Let us now also ignore κz the spatial wave number component along the direction of propagation, by invoking the Markov approximation, which is usually used in the theory of wave propagation in random media (c.f [13].). The Markov approximation implies that the index of refraction is delta-correlated at any pair of points located along the direction of propagation. Under the Markov approximation turbulence is essentially supposed to be layered along the direction of propagation, i.e. the energy transfer process in the inertial sub-range develops only over planes orthogonal to the propagation direction.

3. Propagation of the cross-spectral density matrix of the beam in anisotropic turbulence

Let us now recall that the propagation law for the components W(r1,r2;ω)of the cross-spectral density matrix that characterizes any stochastic electromagnetic beam at a pair of points, say r1 and r2 on propagation in any random linear medium has form [20]:

Wij(r1,r2;ω)=(k2πz)2Wij0(r10,r20;ω)K(r10,r20,r1,r2;ω)d2r10d2r20,(i,j=x,y),
where k=c/ω=2π/λ, superscript “0” denotes points in the source plane, z = 0,
K(r10,r20,r1,r2;ω)=exp[ik(r1r10)2(r2r20)22z]×exp{π2k2z3[(r1r2)2+(r1r2)(r10r20)+(r10r20)2]0κ3Φn(κ)dκ}.
Here Φn(κ) is the power spectrum given in (1). To include the vertical profile we replace C˜n2 with the averaged over the path value,
C˜¯n2=1Hh0h0HC˜n2(h)dh
where C˜n2(h) is the well-known Hufnagel-Valley (H-V) model [21]:
C˜n2(h)=0.00594(vpw27)2(105h)10exp(h1000)+2.7×1016exp(h1500)+C˜n2exp(h100),
and h0,H are the altitudes above the ground where transmitter and receiver are located. The integral in Eq. (5) results in the expression
I=0κ3Φn(κ)dκ=ζ20κ3A(α)C˜¯n2(ζ2κ2+κ02)α2exp(ζ2κ2κm2)dκ=ζ2αA(α)2(α2)C˜¯n2[κ˜m2αβexp(κ02κm2)Γ(2α2,κ02κm2)2κ˜04α].
where β=2κ˜022κ˜m2+ακ˜m2, κ˜02=κ02ζ2,κ˜m2=κm2ζ2 and Γ(,) is the incomplete Gamma function. Anisotropy introduces rescaling of turbulence by the factor ζ2α. Indeed,

I=0κ3Φn(κ)dκ=ζ20κ3A(α)C˜¯n2[ζ2(κ2+κ02ζ2)]α2exp(ζ2κ2κm2)dκ=ζ2α0κ3A(α)C˜¯n2(κ2+κ˜02)α2exp(κ2κ˜m2)dκ=ζ2αA(α)2(α2)C˜¯n2[κ˜m2αβexp(κ02κm2)Γ(2α2,κ02κm2)2κ˜04α].

We will now select the electromagnetic Gaussian-Schell model (EMGSM) [20] as the model source:

Wij0(r10,r20;ω)=BijIiIjexp[(r10)2+(r20)24σ2]exp[(r10r20)22δij2],(i,j=x,y),
where Ix,Iy are the intensities along the x- and y-axis, σis the r.m.s. width of the beam, δijare the r.m.s. correlation coefficients between electric field components i and j, and the coefficients Bij=|Bij|exp[iφij]. It follows from the hermiticity and non-negative definiteness of the cross-spectral density matrix that Bxx=Byy=1,|Bxy|=|Byx|, φxy=φyx.δxy=δyx and max{δxx;δyy}δxymin{δxx/Bxy;δyy/Bxy} [23]. On substituting from Eqs. (5)-(7) into Eq. (4), the elements of the EMGSM beam at distance z from the source can be shown to be
Wij(r1,r2;ω)=BijIiIjΔij2(z)exp[(r1+r2)28σ2Δij2(z)]exp[ik(r22r12)2Rij(z)]×exp{[12Δij2(z)(14σ2+1δij2)+13π2k2zI(1+2Δij2(z))π4k2z4I218σ2Δij2(z)](r1r2)2},
with
Δij2(z)=1+z2k2σ2(14σ2+1δij2)+2π2z3I3σ2,Rij(z)=σ2Δij2(z)zσ2Δij2(z)+(π2z3Iσ2)/3.
The spectral density and the spectral degree of coherence are given by the formulas [20]
S(r;ω)=TrW(r,r;ω),
and
η(r1,r2;ω)=TrW(r1,r2;ω)S(r1;ω)S(r2;ω),
where Tr stands for the trace of the matrix. Further, the spectral degree of polarization is defined as [20]
P(r;ω)=14DetW(r,r;ω)[TrW(r,r;ω)]2=14(WxxWyyWxy2)(Wxx+Wyy)2,
where Det denotes the determinant of the matrix. The parameters of the polarization ellipse relating to the fully polarized portion of the beam, i.e. the orientation angle and the degree of ellipticity were found to be [49]:
θ(r;ω)=12arctan[2ReWxy(r,r;ω)Wxx(r,r;ω)Wyy(r,r;ω)],(π/2<θπ/2)
and
ε(r;ω)=Aminor(r;ω)/Amajor(r;ω),
where the major and minor semi-axes have form

Amajor2(r;ω)=((WxxWyy)2+4|Wxy|2+(WxxWyy)2+4[Re(Wxy)]2)/2,
Aminor2(r;ω)=((WxxWyy)2+4|Wxy|2(WxxWyy)2+4[Re(Wxy)]2)/2.

3. Examples for fully coherent case

We will first examine the limiting case of a coherent, unpolarized (Ax=Ay=1) beam with parametersλ=632.8nm,σ=0.05m(unless other values of parameters are specified). This case is equivalent to a deterministic Gaussian beam with radius σ. We set the following parameters: C˜n2=1014m3α, vpw=21m/s, l0=103m, L0=1m, h0=0,H=30km. The average refractive index structure constant turn out to be C˜¯n2=5.1×1017m3α.

Figure 1 shows the on-axis spectral density of the propagating coherent beam for the chosen combinations of the power law α and anisotropy parameterς. Generally, for higher values of ς the effect of the atmosphere reduces. Regardless of ς the effect on the beam is the greatest for α about 3.1, similarly to the case of isotropic non-Kolmogorov turbulence.

 figure: Fig. 1

Fig. 1 The normalized on-axis spectral density SN of a fully coherent EMGSM beam (r=0): (a) as a function of distance z with different ς and α=3.5; (b) as a function of α and ς at z=30km.

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Figure 2 presents the similar analysis of the r.m.s. beam width calculated by the expression

σij2(z)=σ2Δij2(z),(i,j=x,y).
The r.m.s. width of the propagating beam is shown to be in inverse relation withς, for the fixed values of α. These results agree with those in [32,33].

 figure: Fig. 2

Fig. 2 The beam width of a fully coherent EMGSM source for r=0: (a) as a function of distance z with different ς, (b) as a function of α and ς at z=30km.

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Figure 3 illustrates the changes in the spectral degree of coherence of the propagating beam. In Fig. 3(a) the separation between two points is fixed at rd=0.01m. The degree of coherence decreases with propagation distance with higher rate corresponding to lower values of anisotropy parameter. Figure 3(b) shows the changes continuously for broad range of both α and ς at fixed (large) propagation distance. As for the spectral density and the r.m.s. beam width the degree of coherence is minimal for α between 3.0 and 3.2 for all values of ς. Figure 3(c) demonstrates the evolution of the degree of coherence as a function ς and separation rd between two points, for the fixed value of α=3.5.

 figure: Fig. 3

Fig. 3 The spectral degree of coherence η of fully coherent EMGSM source, (a) as a function of distance z with different ς for r=0, (b) as a function of α and ς at z=30km (c) as a function of the width rd and ς at z=30km.

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4. Examples for partially coherent case

We will now turn to the more general case of a partially coherent beam radiated by source (9). Let the transverse coherence widths be δxx=0.03m,δyy=0.02m, while that other parameters are the same as for fully coherent beam, unless other values of parameters are specified.

Figure 4 shows the changes in the on-axis spectral density for the chosen values of anisotropy and power law parameters. The trends are similar to those in coherent case [see Fig. 1], however Fig. 4(a) implies that in partially coherent case the discrepancy between different beams is much smaller, since the beam behavior is more affected by the source rather than by turbulence. Such small discrepancies in the spectral density of random beams are in line with their propagation in isotropic turbulence investigated in detail previously [22], [28]. It is indeed well known that the behavior of random beams in random media is greatly affected by source correlations, even at large propagation distances, hence the changes in the turbulent statistics can only induce relatively small effects. This is in striking difference with deterministic beams for which a slight change in random media statistics (here the change in anisotropy parameter ζ) lead to substantial changes in beam behavior, in particular, in its spectral density.

 figure: Fig. 4

Fig. 4 The normalized spectral density SN of a typical partially coherent EMGSM beam for r=0, (a) as a function of distance z with different ς, (b) as a function of α and ς at z=30km.

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Figure 5 shows the evolution of the r.m.s. beam width σxx Fig. (a), Fig. (b) and σyy Fig. (c), Fig. (d) of the EMGSM beam. The trends are similar to that in coherent case [see Fig. 3]. However, due to generally different values of r.m.s. correlations δxxand δyythe r.m.s. beam widths are evolving at different rates. As is seen from the figure, anisotropy ς plays a crucial part in predicting the r.m.s. beam width especially at low values ofα.

 figure: Fig. 5

Fig. 5 The r.m.s. beam width of a partially coherent EMGSM beam for r=0: (a) σx and (c)σy as a function of distance z with different ς, (b) σx and (d) σy as a function of α and ς at far field z=30km.

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Figure 6 illustrates the changes in the spectral degree of coherence. In the case of a partially coherent beam it generally grows first due to source correlations and then may start decreasing for sufficiently small anisotropy parameter values [see Fig. 6(a)] due to turbulence effects. Further, the comparison of Figs. 3 and 6, parts Fig. (b) and Fig. (c), implies that the variation of the degree of coherence at fixed large distances and with separation distance are the same in trend for coherent and stochastic beams, while the actual values are lower for the later cases.

 figure: Fig. 6

Fig. 6 The spectral degree of coherenceη, (a) as a function of distance z with different ς for r=0, (b) as a function of α and ς atz=30km(c) as a function of ς and the width r atz=30km.

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Figure 7 shows the evolution of the degree of polarization of the beam on the optical axis. Just like other parameters, it is modulated by turbulence at most for lower values of anisotropy parameter, and for power law parameter close to 3.1. For very high values of anisotropy such as ζ = 10 [see Fig. 7(a)] the degree of polarization is mostly influenced by the source (but not turbulence) and grows monotonically, similarly to its free space behavior. However, for lower anisotropy parameter values (ζ = 1;1.5;2) the atmosphere becomes dominant, making the degree of polarization decrease after reaching a certain maximum.

 figure: Fig. 7

Fig. 7 The degree of polarization P of a partially coherent EMGSM beam for r=0, (a) as a function of distance z with different ς, (b) as a function of α and ς at z=30km.

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Finally, Fig. 8 shows the typical behavior of the polarization ellipse and its parameters for several selected values of anisotropy. Just like the degree of polarization the rest of the polarimetric features are affected less by turbulence with high anisotropy parameter values.

 figure: Fig. 8

Fig. 8 The evolution of polarization ellipse propagating at anisotropic turbulence with different ς for r=0, (a) degree of polarization as a function of distance z with different ς; (b) polarization angle, (c) the degree of ellipticity, (d) polarization ellipse in the far field, z=30km. Other beam parameters are δxy=0.015m, Bxy=0.2eiπ/6 .

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Thus, anisotropic atmosphere does not influence the polarization properties of propagating beams as much as the isotropic one. This conclusion may be of importance for polarization communications and imaging, in which the polarization state is chosen to carry the information instead of (in addition with) conventional average intensity-based transfer.

5. Concluding remarks

In this paper we investigated the effects of anisotropic turbulence on propagation of the electromagnetic Gaussian Schell-model beam. In particular, we have examined how anisotropy affects the spectral density, the states of coherence and polarization of the beam. Our theoretical research was based on a generalized von Karman power spectrum of the index of refraction with a non-Kolmogorov power law and an effective anisotropic parameter, ζ to describe anisotropy along the vertical direction. Anisotropy introduces a rescaling of turbulence, which for the case of the zero inner scale and infinite outer scale is given by the factor ζ2α. The major conclusion that comes out of our analysis is that the high values of the anisotropy parameter lead to suppression of the turbulence effects and enhancement of the source effects on both coherent and random beams. Thus due to anisotropy pertinent to uplink or downlink propagation the turbulence may be effectively weaker compared to conventional cases associated with horizontal links.

Acknowledgments

M. Yao’s research is sponsored by the National Natural Science Foundation of China under Grant No. 11304287; O. Korotkova’s research is supported by US AFOSR (FA9550-12-1-0449) and US ONR (N0018913P1226).

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41. Z. R. Mei and O. Korotkova, “Cosine-Gaussian Schell-model sources,” Opt. Lett. 38(14), 2578–2580 (2013). [CrossRef]   [PubMed]  

42. Z. R. Mei, E. Schchepakina, and O. Korotkova, “Propagation of cosine-Gaussian-correlated Schell-model beams in atmospheric turbulence,” Opt. Express 21(15), 17512–17519 (2013). [CrossRef]   [PubMed]  

43. H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36(20), 4104–4106 (2011). [CrossRef]   [PubMed]  

44. Z. S. Tong and O. Korotkova, “Non-uniformly correlated light beams in uniformly correlated media,” Opt. Lett. 37(15), 3240–3242 (2012). [CrossRef]   [PubMed]  

45. F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007). [CrossRef]   [PubMed]  

46. F. Gori, V. R. Sanchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009). [CrossRef]  

47. W. Lu, L. Liu, J. Sun, Q. Yang, and Y. Zhu, “Change in the degree of coherence of partially coherent electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 271(1), 1–8 (2007). [CrossRef]  

48. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free-space laser beam propagation in non-Kolmogorov turbulence,” Proc. SPIE 6751, 65510E (2007). [CrossRef]  

49. O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246(1-3), 35–43 (2005). [CrossRef]  

References

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  27. G. Wu, H. Guo, S. Yu, and B. Luo, “Spreading and direction of Gaussian-Schell model beam through a non-Kolmogorov turbulence,” Opt. Lett. 35(5), 715–717 (2010).
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  32. I. Toselli, “Introducing the concept of anisotropy at different scales for modeling optical turbulence,” J. Opt. Soc. Am. A 31(8), 1868–1875 (2014).
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  33. I. Toselli, B. Agrawal, and S. Restaino, “Light propagation through anisotropic turbulence,” J. Opt. Soc. Am. A 28(3), 483–488 (2011).
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  34. C. Palma, R. Borghi, and G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125(1-3), 113–121 (1996).
    [Crossref]
  35. F. Gori, M. Santarsiero, and R. Borghi, “Modal expansion for J0-correlated electromagnetic sources,” Opt. Lett. 33(16), 1857–1859 (2008).
    [Crossref] [PubMed]
  36. F. Wang, Y. Cai, and O. Korotkova, “Partially coherent standard and elegant Laguerre-Gaussian beams of all orders,” Opt. Express 17(25), 22366–22379 (2009).
    [Crossref] [PubMed]
  37. S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012).
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    [Crossref] [PubMed]
  41. Z. R. Mei and O. Korotkova, “Cosine-Gaussian Schell-model sources,” Opt. Lett. 38(14), 2578–2580 (2013).
    [Crossref] [PubMed]
  42. Z. R. Mei, E. Schchepakina, and O. Korotkova, “Propagation of cosine-Gaussian-correlated Schell-model beams in atmospheric turbulence,” Opt. Express 21(15), 17512–17519 (2013).
    [Crossref] [PubMed]
  43. H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36(20), 4104–4106 (2011).
    [Crossref] [PubMed]
  44. Z. S. Tong and O. Korotkova, “Non-uniformly correlated light beams in uniformly correlated media,” Opt. Lett. 37(15), 3240–3242 (2012).
    [Crossref] [PubMed]
  45. F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007).
    [Crossref] [PubMed]
  46. F. Gori, V. R. Sanchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
    [Crossref]
  47. W. Lu, L. Liu, J. Sun, Q. Yang, and Y. Zhu, “Change in the degree of coherence of partially coherent electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 271(1), 1–8 (2007).
    [Crossref]
  48. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free-space laser beam propagation in non-Kolmogorov turbulence,” Proc. SPIE 6751, 65510E (2007).
    [Crossref]
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    [Crossref]

2014 (3)

2013 (4)

2012 (3)

2011 (2)

2010 (4)

2009 (4)

F. Wang, Y. Cai, and O. Korotkova, “Partially coherent standard and elegant Laguerre-Gaussian beams of all orders,” Opt. Express 17(25), 22366–22379 (2009).
[Crossref] [PubMed]

D. T. Kyrazis, F. D. Eaton, D. G. Black, W. T. Black, and R. A. Black, “The balloon ring: a high-performance, low-cost instrumentation platform for measuring atmospheric turbulence profiles,” Proc. SPIE 7463, 3–4 (2009).
[Crossref]

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(5), 705–708 (2009).
[Crossref]

F. Gori, V. R. Sanchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[Crossref]

2008 (4)

2007 (3)

W. Lu, L. Liu, J. Sun, Q. Yang, and Y. Zhu, “Change in the degree of coherence of partially coherent electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 271(1), 1–8 (2007).
[Crossref]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free-space laser beam propagation in non-Kolmogorov turbulence,” Proc. SPIE 6751, 65510E (2007).
[Crossref]

F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007).
[Crossref] [PubMed]

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, 6304U (2006).

2005 (2)

L. Biferale and I. Procaccia, “Anisotropy in turbulent flows and in turbulent transport,” Phys. Rep. 414(2-3), 43–164 (2005).
[Crossref]

O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246(1-3), 35–43 (2005).
[Crossref]

1999 (2)

M. S. Belen’kii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect of non-Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396–406 (1999).
[Crossref]

M. S. Belen’kii, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Experimental evidence of the effects of non-Kolmogorov turbulence and anisotropy of turbulence,” Proc. SPIE 3749, 50–51 (1999).
[Crossref]

1998 (1)

F. D. Eaton and G. D. Nastrom, “Preliminary estimates of the vertical profiles of inner and outer scales from White Sands Missile Range,” Radio Sci. 33(4), 895–903 (1998).
[Crossref]

1997 (1)

A. S. Gurvich and V. Kan, “Radio wave fluctuations in satellite–atmosphere–satellite links: estimates from stellar scintillation observations and their comparison with experimental data,” Atmos. Oceanic Phys. 33, 284–292 (1997).

1996 (1)

C. Palma, R. Borghi, and G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125(1-3), 113–121 (1996).
[Crossref]

1995 (2)

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

R. R. Beland, “Some aspects of propagation through weak isotropic non-Kolmogorov turbulence,” Proc. SPIE 2375, 6–16 (1995).
[Crossref]

1994 (3)

D. T. Kyrazis, J. Wissler, D. 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]

F. Dalaudier, C. Sidi, M. Crochet, and J. Vernin, “Direct evidence of ‘sheets’ in the atmospheric temperature Field,” J. Atmos. Sci. 51(2), 237–248 (1994).
[Crossref]

A. I. Kon, “Qualitative theory of amplitude and phase fluctuations in a medium with anisotropic turbulent irregularity,” Waves Rand. Compl. Media 4(3), 297–306 (1994).
[Crossref]

1992 (1)

G. M. Grechko, A. S. Gurvich, V. Kan, S. V. Kireev, and S. A. Savchenko, “Anisotropy of spatial structures in the middle atmosphere,” Adv. Space Res. 12(10), 169–175 (1992).
[Crossref]

1970 (1)

1960 (1)

L. R. Tsvang, “Measurements of the spectrum of temperature fluctuations in the free atmosphere,” Izvestiya Akademii Nauk SSSR, Geofizicheskaya 1, 1117–1120 (1960).

Agrawal, B.

Andrews, L. C.

L. C. Andrews, R. L. Phillips, R. Crabbs, and T. Leclerc, “Deep turbulence propagation of a Gaussian-beam wave in anisotropic non-Kolmogorov turbulence,” Proc. SPIE 8874, 887402 (2013).
[Crossref]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 026003 (2008).
[Crossref]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free-space laser beam propagation in non-Kolmogorov turbulence,” Proc. SPIE 6751, 65510E (2007).
[Crossref]

Barchers, J. D.

M. S. Belen’kii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect of non-Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396–406 (1999).
[Crossref]

Beland, R. R.

R. R. Beland, “Some aspects of propagation through weak isotropic non-Kolmogorov turbulence,” Proc. SPIE 2375, 6–16 (1995).
[Crossref]

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, 6304U (2006).

M. S. Belen’kii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect of non-Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396–406 (1999).
[Crossref]

M. S. Belen’kii, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Experimental evidence of the effects of non-Kolmogorov turbulence and anisotropy of turbulence,” Proc. SPIE 3749, 50–51 (1999).
[Crossref]

Biferale, L.

L. Biferale and I. Procaccia, “Anisotropy in turbulent flows and in turbulent transport,” Phys. Rep. 414(2-3), 43–164 (2005).
[Crossref]

Bishop, K. P.

D. T. Kyrazis, J. Wissler, D. 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]

Black, D. G.

D. T. Kyrazis, F. D. Eaton, D. G. Black, W. T. Black, and R. A. Black, “The balloon ring: a high-performance, low-cost instrumentation platform for measuring atmospheric turbulence profiles,” Proc. SPIE 7463, 3–4 (2009).
[Crossref]

Black, R. A.

D. T. Kyrazis, F. D. Eaton, D. G. Black, W. T. Black, and R. A. Black, “The balloon ring: a high-performance, low-cost instrumentation platform for measuring atmospheric turbulence profiles,” Proc. SPIE 7463, 3–4 (2009).
[Crossref]

Black, W. T.

D. T. Kyrazis, F. D. Eaton, D. G. Black, W. T. Black, and R. A. Black, “The balloon ring: a high-performance, low-cost instrumentation platform for measuring atmospheric turbulence profiles,” Proc. SPIE 7463, 3–4 (2009).
[Crossref]

Borghi, R.

F. Gori, M. Santarsiero, and R. Borghi, “Modal expansion for J0-correlated electromagnetic sources,” Opt. Lett. 33(16), 1857–1859 (2008).
[Crossref] [PubMed]

C. Palma, R. Borghi, and G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125(1-3), 113–121 (1996).
[Crossref]

Brown, J. M.

M. S. Belen’kii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect of non-Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396–406 (1999).
[Crossref]

M. S. Belen’kii, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Experimental evidence of the effects of non-Kolmogorov turbulence and anisotropy of turbulence,” Proc. SPIE 3749, 50–51 (1999).
[Crossref]

Cai, Y.

C. Yan, F. Wang, and Y. Cai, “Propagation of a twist Gaussian–Schell model beam in non-Kolmogorov turbulence,” Opt. Commun. 324, 108–113 (2014).
[Crossref]

F. Wang, Y. Cai, and O. Korotkova, “Partially coherent standard and elegant Laguerre-Gaussian beams of all orders,” Opt. Express 17(25), 22366–22379 (2009).
[Crossref] [PubMed]

Cincotti, G.

C. Palma, R. Borghi, and G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125(1-3), 113–121 (1996).
[Crossref]

Conan, J. M.

Consortini, A.

Crabbs, R.

L. C. Andrews, R. L. Phillips, R. Crabbs, and T. Leclerc, “Deep turbulence propagation of a Gaussian-beam wave in anisotropic non-Kolmogorov turbulence,” Proc. SPIE 8874, 887402 (2013).
[Crossref]

Crochet, M.

F. Dalaudier, C. Sidi, M. Crochet, and J. Vernin, “Direct evidence of ‘sheets’ in the atmospheric temperature Field,” J. Atmos. Sci. 51(2), 237–248 (1994).
[Crossref]

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, 6304U (2006).

Dalaudier, F.

Du, W.

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).
[Crossref] [PubMed]

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(5), 705–708 (2009).
[Crossref]

Eaton, F. D.

D. T. Kyrazis, F. D. Eaton, D. G. Black, W. T. Black, and R. A. Black, “The balloon ring: a high-performance, low-cost instrumentation platform for measuring atmospheric turbulence profiles,” Proc. SPIE 7463, 3–4 (2009).
[Crossref]

F. D. Eaton and G. D. Nastrom, “Preliminary estimates of the vertical profiles of inner and outer scales from White Sands Missile Range,” Radio Sci. 33(4), 895–903 (1998).
[Crossref]

Ferrero, V.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 026003 (2008).
[Crossref]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free-space laser beam propagation in non-Kolmogorov turbulence,” Proc. SPIE 6751, 65510E (2007).
[Crossref]

Fugate, R. Q.

M. S. Belen’kii, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Experimental evidence of the effects of non-Kolmogorov turbulence and anisotropy of turbulence,” Proc. SPIE 3749, 50–51 (1999).
[Crossref]

M. S. Belen’kii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect of non-Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396–406 (1999).
[Crossref]

Golbraikh, E.

Gori, F.

Grechko, G. M.

G. M. Grechko, A. S. Gurvich, V. Kan, S. V. Kireev, and S. A. Savchenko, “Anisotropy of spatial structures in the middle atmosphere,” Adv. Space Res. 12(10), 169–175 (1992).
[Crossref]

Guo, H.

Gurvich, A. S.

A. S. Gurvich and V. Kan, “Radio wave fluctuations in satellite–atmosphere–satellite links: estimates from stellar scintillation observations and their comparison with experimental data,” Atmos. Oceanic Phys. 33, 284–292 (1997).

G. M. Grechko, A. S. Gurvich, V. Kan, S. V. Kireev, and S. A. Savchenko, “Anisotropy of spatial structures in the middle atmosphere,” Adv. Space Res. 12(10), 169–175 (1992).
[Crossref]

Han, Q.

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, 6304U (2006).

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(5), 705–708 (2009).
[Crossref]

Kan, V.

A. S. Gurvich and V. Kan, “Radio wave fluctuations in satellite–atmosphere–satellite links: estimates from stellar scintillation observations and their comparison with experimental data,” Atmos. Oceanic Phys. 33, 284–292 (1997).

G. M. Grechko, A. S. Gurvich, V. Kan, S. V. Kireev, and S. A. Savchenko, “Anisotropy of spatial structures in the middle atmosphere,” Adv. Space Res. 12(10), 169–175 (1992).
[Crossref]

Karis, S. J.

M. S. Belen’kii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect of non-Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396–406 (1999).
[Crossref]

M. S. Belen’kii, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Experimental evidence of the effects of non-Kolmogorov turbulence and anisotropy of turbulence,” Proc. SPIE 3749, 50–51 (1999).
[Crossref]

Keating, D. D. B.

D. T. Kyrazis, J. Wissler, D. 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]

Kireev, S. V.

G. M. Grechko, A. S. Gurvich, V. Kan, S. V. Kireev, and S. A. Savchenko, “Anisotropy of spatial structures in the middle atmosphere,” Adv. Space Res. 12(10), 169–175 (1992).
[Crossref]

Kon, A. I.

A. I. Kon, “Qualitative theory of amplitude and phase fluctuations in a medium with anisotropic turbulent irregularity,” Waves Rand. Compl. Media 4(3), 297–306 (1994).
[Crossref]

Kopeika, N. S.

Korotkova, O.

O. Korotkova, “Random sources for rectangular far fields,” Opt. Lett. 39(1), 64–67 (2014).
[Crossref] [PubMed]

Z. R. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013).
[Crossref] [PubMed]

Z. R. Mei, E. Schchepakina, and O. Korotkova, “Propagation of cosine-Gaussian-correlated Schell-model beams in atmospheric turbulence,” Opt. Express 21(15), 17512–17519 (2013).
[Crossref] [PubMed]

Z. R. Mei and O. Korotkova, “Cosine-Gaussian Schell-model sources,” Opt. Lett. 38(14), 2578–2580 (2013).
[Crossref] [PubMed]

Z. S. Tong and O. Korotkova, “Non-uniformly correlated light beams in uniformly correlated media,” Opt. Lett. 37(15), 3240–3242 (2012).
[Crossref] [PubMed]

S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012).
[Crossref] [PubMed]

O. Korotkova, S. Sahin, and E. Shchepakina, “Multi-Gaussian Schell-model beams,” J. Opt. Soc. Am. A 29(10), 2159–2164 (2012).
[Crossref] [PubMed]

E. Shchepakina and O. Korotkova, “Second-order statistics of stochastic electromagnetic beams propagating through non-Kolmogorov turbulence,” Opt. Express 18(10), 10650–10658 (2010).
[Crossref] [PubMed]

O. Korotkova and E. Shchepakina, “Color changes in stochastic light fields propagating in non-Kolmogorov turbulence,” Opt. Lett. 35(22), 3772–3774 (2010).
[Crossref] [PubMed]

F. Wang, Y. Cai, and O. Korotkova, “Partially coherent standard and elegant Laguerre-Gaussian beams of all orders,” Opt. Express 17(25), 22366–22379 (2009).
[Crossref] [PubMed]

O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246(1-3), 35–43 (2005).
[Crossref]

Kyrazis, D. T.

D. T. Kyrazis, F. D. Eaton, D. G. Black, W. T. Black, and R. A. Black, “The balloon ring: a high-performance, low-cost instrumentation platform for measuring atmospheric turbulence profiles,” Proc. SPIE 7463, 3–4 (2009).
[Crossref]

D. T. Kyrazis, J. Wissler, D. 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]

Lajunen, H.

Leclerc, T.

L. C. Andrews, R. L. Phillips, R. Crabbs, and T. Leclerc, “Deep turbulence propagation of a Gaussian-beam wave in anisotropic non-Kolmogorov turbulence,” Proc. SPIE 8874, 887402 (2013).
[Crossref]

Liu, L.

W. Lu, L. Liu, J. Sun, Q. Yang, and Y. Zhu, “Change in the degree of coherence of partially coherent electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 271(1), 1–8 (2007).
[Crossref]

Lu, W.

W. Lu, L. Liu, J. Sun, Q. Yang, and Y. Zhu, “Change in the degree of coherence of partially coherent electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 271(1), 1–8 (2007).
[Crossref]

Luo, B.

Ma, J.

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).
[Crossref] [PubMed]

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(5), 705–708 (2009).
[Crossref]

Mei, Z. R.

Michau, V.

Nastrom, G. D.

F. D. Eaton and G. D. Nastrom, “Preliminary estimates of the vertical profiles of inner and outer scales from White Sands Missile Range,” Radio Sci. 33(4), 895–903 (1998).
[Crossref]

Osmon, C. L.

M. S. Belen’kii, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Experimental evidence of the effects of non-Kolmogorov turbulence and anisotropy of turbulence,” Proc. SPIE 3749, 50–51 (1999).
[Crossref]

M. S. Belen’kii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect of non-Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396–406 (1999).
[Crossref]

Palma, C.

C. Palma, R. Borghi, and G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125(1-3), 113–121 (1996).
[Crossref]

Phillips, R. L.

L. C. Andrews, R. L. Phillips, R. Crabbs, and T. Leclerc, “Deep turbulence propagation of a Gaussian-beam wave in anisotropic non-Kolmogorov turbulence,” Proc. SPIE 8874, 887402 (2013).
[Crossref]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 026003 (2008).
[Crossref]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free-space laser beam propagation in non-Kolmogorov turbulence,” Proc. SPIE 6751, 65510E (2007).
[Crossref]

Preble, A. J.

D. T. Kyrazis, J. Wissler, D. 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]

Procaccia, I.

L. Biferale and I. Procaccia, “Anisotropy in turbulent flows and in turbulent transport,” Phys. Rep. 414(2-3), 43–164 (2005).
[Crossref]

Renard, J. B.

Restaino, S.

Robert, C.

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]

Ronchi, L.

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, 6304U (2006).

Saastamoinen, T.

Sahin, S.

Sanchez, V. R.

F. Gori, V. R. Sanchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[Crossref]

Santarsiero, M.

Savchenko, S. A.

G. M. Grechko, A. S. Gurvich, V. Kan, S. V. Kireev, and S. A. Savchenko, “Anisotropy of spatial structures in the middle atmosphere,” Adv. Space Res. 12(10), 169–175 (1992).
[Crossref]

Schchepakina, E.

Shchepakina, E.

Shirai, T.

F. Gori, V. R. Sanchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[Crossref]

Sidi, C.

F. Dalaudier, C. Sidi, M. Crochet, and J. Vernin, “Direct evidence of ‘sheets’ in the atmospheric temperature Field,” J. Atmos. Sci. 51(2), 237–248 (1994).
[Crossref]

Stefanutti, L.

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]

Sun, J.

W. Lu, L. Liu, J. Sun, Q. Yang, and Y. Zhu, “Change in the degree of coherence of partially coherent electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 271(1), 1–8 (2007).
[Crossref]

Tan, L.

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).
[Crossref] [PubMed]

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(5), 705–708 (2009).
[Crossref]

Tong, Z. S.

Toselli, I.

I. Toselli, “Introducing the concept of anisotropy at different scales for modeling optical turbulence,” J. Opt. Soc. Am. A 31(8), 1868–1875 (2014).
[Crossref] [PubMed]

I. Toselli, B. Agrawal, and S. Restaino, “Light propagation through anisotropic turbulence,” J. Opt. Soc. Am. A 28(3), 483–488 (2011).
[Crossref] [PubMed]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 026003 (2008).
[Crossref]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free-space laser beam propagation in non-Kolmogorov turbulence,” Proc. SPIE 6751, 65510E (2007).
[Crossref]

Tsvang, L. R.

L. R. Tsvang, “Measurements of the spectrum of temperature fluctuations in the free atmosphere,” Izvestiya Akademii Nauk SSSR, Geofizicheskaya 1, 1117–1120 (1960).

Vernin, J.

F. Dalaudier, C. Sidi, M. Crochet, and J. Vernin, “Direct evidence of ‘sheets’ in the atmospheric temperature Field,” J. Atmos. Sci. 51(2), 237–248 (1994).
[Crossref]

Wang, F.

C. Yan, F. Wang, and Y. Cai, “Propagation of a twist Gaussian–Schell model beam in non-Kolmogorov turbulence,” Opt. Commun. 324, 108–113 (2014).
[Crossref]

F. Wang, Y. Cai, and O. Korotkova, “Partially coherent standard and elegant Laguerre-Gaussian beams of all orders,” Opt. Express 17(25), 22366–22379 (2009).
[Crossref] [PubMed]

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]

Wissler, J.

D. T. Kyrazis, J. Wissler, D. 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]

Wolf, E.

O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246(1-3), 35–43 (2005).
[Crossref]

Wu, G.

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(5), 705–708 (2009).
[Crossref]

Yan, C.

C. Yan, F. Wang, and Y. Cai, “Propagation of a twist Gaussian–Schell model beam in non-Kolmogorov turbulence,” Opt. Commun. 324, 108–113 (2014).
[Crossref]

Yang, Q.

W. Lu, L. Liu, J. Sun, Q. Yang, and Y. Zhu, “Change in the degree of coherence of partially coherent electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 271(1), 1–8 (2007).
[Crossref]

Yu, S.

Zhu, Y.

W. Lu, L. Liu, J. Sun, Q. Yang, and Y. Zhu, “Change in the degree of coherence of partially coherent electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 271(1), 1–8 (2007).
[Crossref]

Zilberman, A.

Adv. Space Res. (1)

G. M. Grechko, A. S. Gurvich, V. Kan, S. V. Kireev, and S. A. Savchenko, “Anisotropy of spatial structures in the middle atmosphere,” Adv. Space Res. 12(10), 169–175 (1992).
[Crossref]

Appl. Opt. (2)

Atmos. Oceanic Phys. (1)

A. S. Gurvich and V. Kan, “Radio wave fluctuations in satellite–atmosphere–satellite links: estimates from stellar scintillation observations and their comparison with experimental data,” Atmos. Oceanic Phys. 33, 284–292 (1997).

Izvestiya Akademii Nauk SSSR, Geofizicheskaya (1)

L. R. Tsvang, “Measurements of the spectrum of temperature fluctuations in the free atmosphere,” Izvestiya Akademii Nauk SSSR, Geofizicheskaya 1, 1117–1120 (1960).

J. Atmos. Sci. (1)

F. Dalaudier, C. Sidi, M. Crochet, and J. Vernin, “Direct evidence of ‘sheets’ in the atmospheric temperature Field,” J. Atmos. Sci. 51(2), 237–248 (1994).
[Crossref]

J. Opt. A, Pure Appl. Opt. (1)

F. Gori, V. R. Sanchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 085706 (2009).
[Crossref]

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

Opt. Commun. (5)

C. Palma, R. Borghi, and G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125(1-3), 113–121 (1996).
[Crossref]

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(5), 705–708 (2009).
[Crossref]

C. Yan, F. Wang, and Y. Cai, “Propagation of a twist Gaussian–Schell model beam in non-Kolmogorov turbulence,” Opt. Commun. 324, 108–113 (2014).
[Crossref]

W. Lu, L. Liu, J. Sun, Q. Yang, and Y. Zhu, “Change in the degree of coherence of partially coherent electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 271(1), 1–8 (2007).
[Crossref]

O. Korotkova and E. Wolf, “Changes in the state of polarization of a random electromagnetic beam on propagation,” Opt. Commun. 246(1-3), 35–43 (2005).
[Crossref]

Opt. Eng. (1)

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47(2), 026003 (2008).
[Crossref]

Opt. Express (4)

Opt. Lett. (10)

G. Wu, H. Guo, S. Yu, and B. Luo, “Spreading and direction of Gaussian-Schell model beam through a non-Kolmogorov turbulence,” Opt. Lett. 35(5), 715–717 (2010).
[Crossref] [PubMed]

O. Korotkova and E. Shchepakina, “Color changes in stochastic light fields propagating in non-Kolmogorov turbulence,” Opt. Lett. 35(22), 3772–3774 (2010).
[Crossref] [PubMed]

F. Gori, M. Santarsiero, and R. Borghi, “Modal expansion for J0-correlated electromagnetic sources,” Opt. Lett. 33(16), 1857–1859 (2008).
[Crossref] [PubMed]

O. Korotkova, “Random sources for rectangular far fields,” Opt. Lett. 39(1), 64–67 (2014).
[Crossref] [PubMed]

Z. R. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013).
[Crossref] [PubMed]

Z. R. Mei and O. Korotkova, “Cosine-Gaussian Schell-model sources,” Opt. Lett. 38(14), 2578–2580 (2013).
[Crossref] [PubMed]

H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36(20), 4104–4106 (2011).
[Crossref] [PubMed]

Z. S. Tong and O. Korotkova, “Non-uniformly correlated light beams in uniformly correlated media,” Opt. Lett. 37(15), 3240–3242 (2012).
[Crossref] [PubMed]

F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007).
[Crossref] [PubMed]

S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012).
[Crossref] [PubMed]

Phys. Rep. (1)

L. Biferale and I. Procaccia, “Anisotropy in turbulent flows and in turbulent transport,” Phys. Rep. 414(2-3), 43–164 (2005).
[Crossref]

Proc. SPIE (9)

M. S. Belen’kii, J. D. Barchers, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Preliminary experimental evidence of anisotropy of turbulence and the effect of non-Kolmogorov turbulence on wavefront tilt statistics,” Proc. SPIE 3762, 396–406 (1999).
[Crossref]

M. S. Belen’kii, S. J. Karis, C. L. Osmon, J. M. Brown, and R. Q. Fugate, “Experimental evidence of the effects of non-Kolmogorov turbulence and anisotropy of turbulence,” Proc. SPIE 3749, 50–51 (1999).
[Crossref]

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

R. R. Beland, “Some aspects of propagation through weak isotropic non-Kolmogorov turbulence,” Proc. SPIE 2375, 6–16 (1995).
[Crossref]

D. T. Kyrazis, J. Wissler, D. 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]

D. T. Kyrazis, F. D. Eaton, D. G. Black, W. T. Black, and R. A. Black, “The balloon ring: a high-performance, low-cost instrumentation platform for measuring atmospheric turbulence profiles,” Proc. SPIE 7463, 3–4 (2009).
[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, 6304U (2006).

L. C. Andrews, R. L. Phillips, R. Crabbs, and T. Leclerc, “Deep turbulence propagation of a Gaussian-beam wave in anisotropic non-Kolmogorov turbulence,” Proc. SPIE 8874, 887402 (2013).
[Crossref]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free-space laser beam propagation in non-Kolmogorov turbulence,” Proc. SPIE 6751, 65510E (2007).
[Crossref]

Radio Sci. (1)

F. D. Eaton and G. D. Nastrom, “Preliminary estimates of the vertical profiles of inner and outer scales from White Sands Missile Range,” Radio Sci. 33(4), 895–903 (1998).
[Crossref]

Waves Rand. Compl. Media (1)

A. I. Kon, “Qualitative theory of amplitude and phase fluctuations in a medium with anisotropic turbulent irregularity,” Waves Rand. Compl. Media 4(3), 297–306 (1994).
[Crossref]

Other (6)

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

E. Wolf, Introduction to the Theories of Coherence and Polarization of Light (Cambridge University, 2007).

L. C. Andrews and R. L. Phillips, Laser Beam Propagation in Random Media, 2nd ed. (SPIE 2005).
[Crossref]

O. Korotkova, Propagation of Partially Coherent Beams in Turbulent Atmosphere (WDM, 2009).

O. Korotkova, Random Beams: Theory and Applications (CRC, 2013).

Cited By

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Figures (8)

Fig. 1
Fig. 1 The normalized on-axis spectral density S N of a fully coherent EMGSM beam ( r=0 ): (a) as a function of distance z with different ς and α=3.5 ; (b) as a function of α and ς at z=30km .
Fig. 2
Fig. 2 The beam width of a fully coherent EMGSM source for r=0 : (a) as a function of distance z with different ς, (b) as a function of α and ς at z=30km .
Fig. 3
Fig. 3 The spectral degree of coherence η of fully coherent EMGSM source, (a) as a function of distance z with different ς for r=0 , (b) as a function of α and ς at z=30km (c) as a function of the width r d and ς at z=30km .
Fig. 4
Fig. 4 The normalized spectral density S N of a typical partially coherent EMGSM beam for r=0 , (a) as a function of distance z with different ς, (b) as a function of α and ς at z=30km .
Fig. 5
Fig. 5 The r.m.s. beam width of a partially coherent EMGSM beam for r=0 : (a) σ x and (c) σ y as a function of distance z with different ς, (b) σ x and (d) σ y as a function of α and ς at far field z=30km .
Fig. 6
Fig. 6 The spectral degree of coherenceη, (a) as a function of distance z with different ς for r=0 , (b) as a function of α and ς at z=30km (c) as a function of ς and the width r at z=30km .
Fig. 7
Fig. 7 The degree of polarization P of a partially coherent EMGSM beam for r=0 , (a) as a function of distance z with different ς, (b) as a function of α and ς at z=30km .
Fig. 8
Fig. 8 The evolution of polarization ellipse propagating at anisotropic turbulence with different ς for r=0 , (a) degree of polarization as a function of distance z with different ς; (b) polarization angle, (c) the degree of ellipticity, (d) polarization ellipse in the far field, z=30km . Other beam parameters are δ xy =0.015m , B xy =0.2 e iπ/6 .

Equations (20)

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

Φ n ( κ,α )=A( α ) C ˜ n 2 ζ 2 ( ζ 2 κ xy 2 + κ z 2 + κ 0 2 ) α 2 exp( ζ 2 κ xy 2 + κ z 2 κ m 2 ),κ>0,3<α<5,
C( α )= { πA( α )Γ( 3 2 α 2 )( 3α 3 ) } 1 α5 ,3<α<5,
A( α )= Γ( α1 ) 4 π 2 cos( α π 2 ),3<α<5,
W ij ( r 1 , r 2 ;ω )= ( k 2πz ) 2 W ij 0 ( r 1 0 , r 2 0 ;ω )K( r 1 0 , r 2 0 , r 1 , r 2 ;ω ) d 2 r 1 0 d 2 r 2 0 ,(i,j=x,y),
K( r 1 0 , r 2 0 , r 1 , r 2 ;ω )=exp[ ik ( r 1 r 1 0 ) 2 ( r 2 r 2 0 ) 2 2z ] ×exp{ π 2 k 2 z 3 [ ( r 1 r 2 ) 2 +( r 1 r 2 )( r 1 0 r 2 0 )+ ( r 1 0 r 2 0 ) 2 ] 0 κ 3 Φ n ( κ )dκ }.
C ˜ ¯ n 2 = 1 H h 0 h 0 H C ˜ n 2 ( h ) dh
C ˜ n 2 ( h )=0.00594 ( v pw 27 ) 2 ( 10 5 h ) 10 exp( h 1000 )+2.7× 10 16 exp( h 1500 )+ C ˜ n 2 exp( h 100 ),
I= 0 κ 3 Φ n ( κ ) dκ= ζ 2 0 κ 3 A( α ) C ˜ ¯ n 2 ( ζ 2 κ 2 + κ 0 2 ) α 2 exp( ζ 2 κ 2 κ m 2 )dκ = ζ 2α A( α ) 2( α2 ) C ˜ ¯ n 2 [ κ ˜ m 2α βexp( κ 0 2 κ m 2 )Γ( 2 α 2 , κ 0 2 κ m 2 )2 κ ˜ 0 4α ].
I= 0 κ 3 Φ n ( κ ) dκ= ζ 2 0 κ 3 A( α ) C ˜ ¯ n 2 [ ζ 2 ( κ 2 + κ 0 2 ζ 2 ) ] α 2 exp( ζ 2 κ 2 κ m 2 )dκ = ζ 2α 0 κ 3 A( α ) C ˜ ¯ n 2 ( κ 2 + κ ˜ 0 2 ) α 2 exp( κ 2 κ ˜ m 2 )dκ = ζ 2α A( α ) 2( α2 ) C ˜ ¯ n 2 [ κ ˜ m 2α βexp( κ 0 2 κ m 2 )Γ( 2 α 2 , κ 0 2 κ m 2 )2 κ ˜ 0 4α ].
W ij 0 ( r 1 0 , r 2 0 ;ω )= B ij I i I j exp[ ( r 1 0 ) 2 + ( r 2 0 ) 2 4 σ 2 ]exp[ ( r 1 0 r 2 0 ) 2 2 δ ij 2 ],(i,j=x,y),
W ij ( r 1 , r 2 ;ω )= B ij I i I j Δ ij 2 ( z ) exp[ ( r 1 + r 2 ) 2 8 σ 2 Δ ij 2 ( z ) ]exp[ ik( r 2 2 r 1 2 ) 2 R ij ( z ) ] ×exp{ [ 1 2 Δ ij 2 ( z ) ( 1 4 σ 2 + 1 δ ij 2 )+ 1 3 π 2 k 2 zI( 1+ 2 Δ ij 2 ( z ) ) π 4 k 2 z 4 I 2 18 σ 2 Δ ij 2 ( z ) ] ( r 1 r 2 ) 2 },
Δ ij 2 ( z )=1+ z 2 k 2 σ 2 ( 1 4 σ 2 + 1 δ ij 2 )+ 2 π 2 z 3 I 3 σ 2 , R ij ( z )= σ 2 Δ ij 2 ( z )z σ 2 Δ ij 2 ( z )+ ( π 2 z 3 I σ 2 ) /3 .
S( r;ω )=TrW( r,r;ω ),
η( r 1 , r 2 ;ω )= TrW( r 1 , r 2 ;ω ) S( r 1 ;ω ) S( r 2 ;ω ) ,
P( r;ω )= 1 4DetW( r,r;ω ) [ TrW( r,r;ω ) ] 2 = 1 4( W xx W yy W xy 2 ) ( W xx + W yy ) 2 ,
θ( r;ω )= 1 2 arctan[ 2Re W xy ( r,r;ω ) W xx ( r,r;ω ) W yy ( r,r;ω ) ],( π/2<θπ/2 )
ε( r;ω )= A minor ( r;ω )/ A major ( r;ω ),
A major 2 ( r;ω )= ( ( W xx W yy ) 2 +4 | W xy | 2 + ( W xx W yy ) 2 +4 [ Re( W xy ) ] 2 ) /2 ,
A minor 2 ( r;ω )= ( ( W xx W yy ) 2 +4 | W xy | 2 ( W xx W yy ) 2 +4 [ Re( W xy ) ] 2 ) /2 .
σ ij 2 ( z )= σ 2 Δ ij 2 ( z ),(i,j=x,y).

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