Abstract

Applying the angular spectrum theory, we derive the expression of a new Hermite-Gaussian (HG) vortex beam. Based on the new Hermite-Gaussian (HG) vortex beam, we establish the model of the received probability density of orbital angular momentum (OAM) modes of this beam propagating through a turbulent ocean of anisotropy. By numerical simulation, we investigate the influence of oceanic turbulence and beam parameters on the received probability density of signal OAM modes and crosstalk OAM modes of the HG vortex beam. The results show that the influence of oceanic turbulence of anisotropy on the received probability of signal OAM modes is smaller than isotropic oceanic turbulence under the same condition, and the effect of salinity fluctuation on the received probability of the signal OAM modes is larger than the effect of temperature fluctuation. In the strong dissipation of kinetic energy per unit mass of fluid and the weak dissipation rate of temperature variance, we can decrease the effects of turbulence on the received probability of signal OAM modes by selecting a long wavelength and a larger transverse size of the HG vortex beam in the source’s plane. In long distance propagation, the HG vortex beam is superior to the Laguerre-Gaussian beam for resisting the destruction of oceanic turbulence.

© 2017 Optical Society of America

1. Introduction

In recent years, optical beams with orbital angular momentum (OAM) propagating in random media have attracted much attention due to the applications in optical wireless communication. The main motivation is that beams carrying OAM can realize arbitrary base-N quantum digits in principle [1,2], which offers the additional degrees of freedom for information coding and increases information capacity of the communication link. Due to the presence of various constituents in the ocean and the fluctuations of the refractive index of water, the propagation of optical signal through an oceanic medium will suffer attenuation and wavefront distortion [3,4]. It is well known that oceanic turbulence will distort OAM modes of optical beams, which introduces intermodal crosstalk between channels. After Nikishov et al. [5] presented the power spectrum of oceanic turbulence, there are great attentions paid to understand the laws of OAM modes propagation in the oceanic turbulence [6–10]. Huang et al. [9] analyzed the influence of beam parameters, the rate of dissipation of turbulent and temperature salinity fluctuations on the average intensity of Gaussian Schell-model vortex beam propagating through oceanic turbulence, and showed that the partially coherent vortex beams have robust turbulent resistance. Liu et al. [10] studied the evolution properties of partially coherent flat-topped vortex hollow beams propagating through oceanic turbulence, and revealed that this beam evolves into the Gaussian-like beam for the long distance propagation and this beam with higher order M will lose its initial dark hollow center slower. The effects of oceanic turbulence and light source parameters on the received probability of partially coherent Laguerre–Gaussian (LG) vortex beam was investigated [6], and showed that the partially coherent vortex beam provides worse performance than fully coherent vortex beam. The channel capacity of Bessel–Gauss Beams propagating through turbulent ocean was investigated [7], which revealed that BG beams are better than LG beams to mitigate the effects of oceanic turbulence because of the non-diffraction and self-healing characteristics of BG beams.

The experimental and theoretical results demonstrated that turbulence can be anisotropic [11–17]. Due to the earth's rotation, the turbulence of ocean will also be anisotropic [18,19]. The average polarizability of Gaussian Schell-model quantized beams and the spreading and wandering of Gaussian–Schell model laser beams in anisotropic oceanic turbulence were studied respectively [18,19]. However, the energy distribution of OAM modes of LG beams decreases with the increasing quantum number of OAM modes, and the crosstalk between OAM modes increases with increasing the quantum number of OAM modes [2,7]. We cannot make use of the high order OAM modes of LG beams to improve the capacity of the free-space optical links. To improve capacity of the free-space optical links with the high order OAM modes, a more suitable vortex beam should be found. To the best of our knowledge, the transmission properties of Hermite-Gaussian (HG) vortex beams carrying OAM in anisotropic turbulent ocean have not been reported yet.

In this paper, we first develop a new angular spectrum of the vortex beam and derive a new HG vortex beam based on the angular spectrum theory. Then, we study the propagation of the new HG vortex beam in anisotropic turbulent ocean, and establish the model of the received probability distribution of OAM modes. In the end, we discuss the influences of the HG vortex beam parameters and the anisotropic turbulent ocean on probability density of signal OAM modes and crosstalk OAM modes in weak horizontal oceanic turbulent channels.

2. Anisotropic oceanic turbulence

In this paper, we study the HG vortex beam propagation in the turbulence of the pure seawater [20]. Due to the earth’s rotation, the turbulence of the ocean will be anisotropic [11]. For the sake of simplicity, here we employ the method that the anisotropic power spectrum has been proposed in atmospheric turbulence [15–17] and consider the turbulent anisotropy only existing along the propagation direction of the beam as the discussion of Ref [18]. We can present the anisotropic power spectrum of oceanic turbulence by an effective anisotropic factor ζ. The anisotropic spectrum of oceanic turbulence can be expressed as

ϕ(κ,ζ)=0.388Cm2ζ2κ11/3[1+2.35(κη)2/3]f(κ,ζ,ϖ),
where κ=κz2+ζ2κρ2is the spatial frequency of turbulent fluctuations, κρ=κx2+κy2. Cm2=108ε1/3χt is the temperature structure constant for given ϖ [20] that characters the temperature fluctuation strength of oceanic turbulence, ε is the rate of dissipation of kinetic energy per unit mass of fluid ranging from 1010m2/s3 to 101m2/s3, χt is the dissipation rate of temperature variance and has the range from1010K2/s to 102K2/s, η is the inner scale of oceanic turbulence, f(κ,ϖ)=[exp(ATδ)+ϖ2exp(ASδ)2ϖ1exp(ATSδ)], AT=1.863×102, AS=1.9×104, ATS=9.41×103, ϖ is the ratio of temperature and salinity contributions to the refractive index spectrum, which in the seawaters can vary in the interval[5;0], with 5 and 0 the corresponding to dominating temperature-induced and salinity-induced optical turbulence, respectively; δ=8.284(κη)4/3+12.978(κη)2.

3. Propagation of HG vortex beams in the absence of turbulence

The propagation of a monochromatic scalar light field u(x,y,z;ω) in free space is represented by the angular spectrum for the half-space z>0 [21,22]:

u(x,y,z;ω)=exp(ikz)4π2a(kx,ky;ω)exp[i(kxx+kyy)]exp[iz2k(kx2+ky2)]dkxdky,
where ω is the angular frequency, k=2π/λ=ω/c is the wavenumber in free space, λ is the wavelength, kxand ky represent the components of the wave vector k along the x axis and y axis respectively, z is transmission distance, a(kx,ky;ω) represents the angular spectrum.

For the lowest-order Gaussian beam field, the product of the divergence Δθ(z;ω) and transverse spread Δω(z;ω) reaches a minimum and can be represented by [22]:

Δθ(z;ω)Δω(z;ω)min.

The expressions of the divergence and transverse spread for the xy symmetry can be defined as

Δθ(z;ω)=14π2(kx2+ky2)|a(kx,ky;ω)|2dkxdky=14π2kx2|a(kx,ky;ω)|2dkxky2|a(kx,ky;ω)|2dky,
Δω(z;ω)=14π2(|a(kx,ky;ω)kx|2+|a(kx,ky;ω)ky|2)dkxdky,=14π2|a(kx,ky;ω)kx|2dkx|a(kx,ky;ω)ky|2dky
where represents the partial derivative.

The product of the divergence and transverse spread in x or y direction meets the minimum respectively. In x direction, the product can be expressed by [22]:

14π2kx2|a(kx,ky;ω)|2dkx|a(kx,ky;ω)kx|2dkmin.

Applying the Cauchy-Schwartz inequality

|f(x)|2dx|g(x)|2dx|f*(x)g(x)dx|2,
where f() and g() stand for the arbitrary functions, the superscript * represents the complex conjugate.

Equation (7) can reach its minimum in the condition that kx2|a(kx,ky;ω)|2 and |a(kx,ky;ω)kx|2are proportional to each other. The angular spectrum for the Gaussian beams can be represented by

a(kx,ky;ω)=exp[w024(kx2+ky2)].

It is known that the angular spectrum of the HG beam is given by [21,22]

a(kx,ky;ω)=kxnkynexp[w024(kx2+ky2)].

Inserting the vortex factor (kxiky)m into the angular spectrum of the HG beam, we can obtain a new angular spectrum. We call this new angular spectrum as the angular spectrum of the HG vortex beam and express it as

a(kx,ky;ω)=kxnkyn(kxiky)mexp[w024(kx2+ky2)],
where n and m are integers.

Substituting Eq. (10) into Eq. (2), the propagation of the HG vortex beam in free space of absence turbulence can be investigated by modulating the angular spectrum

u(x,y,z;ω)=exp(ikz)4π2kxnkyn(kxiky)mexp[ϒ(kx,ky)]dkxdky,
where
exp[ϒ(kx,ky)]=exp[w024(1+izζ)(kx2+ky2)]exp[i(kxx+kyy)],
and zζ=λz/(πw02), w0 is the initial radius of beams in the source plane.

According to these relations

nexp[i(kxx+kyy)]xn=(ikx)nexp[i(kxx+kyy)],
nexp[i(kxx+kyy)]yn=(iky)nexp[i(kxx+kyy)],
(xiy)mexp[i(kxx+kyy)]=im(kxiky)mexp[i(kxx+kyy)].
Equation (11) can be rewritten by

u(x,y,z;ω)=exp(ikz)4π22nxnyn(xiy)mexp[ϒ(kx,ky)]dkxdky.

By setting the new variablesγa=x+iy and γb=xiy, the partial derivative becomes xiy=γa. Therefore, Eq. (16) can be rewritten by

u(x,y,z;ω)=exp(ikz)4π22nxnynγamexp[ϒ(kx,ky)]dkxdky,

With the help of the integral expression [21–23]

12πexp(iaxb22x2)dx=1b2πexp[a22b2],
the HG vortex beam is given by

um(x,y,z;ω)=exp(ikz)1[w02(1+izζ)]m+12nxnynγbmexp[γaγbw02(1+izζ)].

Neglecting the constant factor, we can obtain the HG vortex beams by using the following mathematical formulae γaγb=x2+y2 and (x+y)m=u=0mm!(mu)!u!xmuyu as

um(x,y,z;ω)=exp(imφ)exp(ikz)[w02(1+izζ)]m+1u=0m/2(m/2)!(m/2u)!u!HVnm2u(x)HVn2u(y),
where φ=arctan(y/x),HVnm2u(x)=dndxnxm2uexp[x2w02(1+izζ)], and HVn2u(y)=dndyny2u ×exp[y2w02(1+izζ)]. The Fig. 1 presents the intensity distribution for HG vortex beams um(x,y,z;ω) in the planes z=0.

 

Fig. 1 Intensity profiles of HG vortex beams in the absence of turbulence (a) n=2,m=2, (b) n=2,m=4,(c) n=2,m=6. When z=0 in the plane, the intensity of the HG vortex beam decreases with deviating from the optical axis center. In addition, the intensity forms annular shapes along the optical axis center with increasing quantum number m.

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In particular, when m=0, the new HG vortex beam can degrade into HG beam as

um(x,y,z;ω)=1[w02(1+izζ)]n+1Hn(x/w02(1+izζ))×Hn(yw02(1+izζ))exp[ikzx2+y2w02(1+izζ)],
where Hn() represents Hermite polynomials.

Moreover, let n=0 and r2=x2+y2, the new HG vortex beam can be reduced as LGm0 beam [24] and can be given by

um(x,y,z;ω)=exp(imφ)exp(ikz)[w02(1+izζ)]m+1(r)mexp[r2w02(1+izζ)]Lm0[r2w02(1+izζ)],
where Lm0() represents Laguerre polynomials.

4. Probability density of OAM modes in oceanic turbulence

The HG vortex beam u(r,φ,z) is a superposition of eigenstates and the conditional probability of obtaining a measurement of the OAM of a photonlz=l is obtained by summing the probabilities associated with that eigenvalue [2],

p(l|u)=l|al(r,z)|2,
where l is the quantum number of the OAM modes of beams in turbulence, and al(r,z) is the superposition coefficients.

The aberrations are random, so by taking the ensemble average, the probability that a measurement of the OAM will yield a value lz=l is found as follows,

p(l/m)=p(l|u)=m|al(r,z)|2,
where denotes average over the ensemble of oceanic turbulence.

By the superposition theory of plane waves with phase exp(ilφ), the al(r,z) can be written as

al(r,z)=12π02πu(r,φ,z)exp(ilφ)dφ.
where u(r,φ,z) represents the field amplitude of the HG vortex beam propagation in oceanic turbulence.

The complex amplitude of the HG vortex beam is given by [25]

u(r,φ,z)=um(r,φ,z)exp[iψ(r,φ)],
where ψ(r,φ) is the phase distortion caused by oceanic turbulence.

The ensemble average of the superposition coefficients al(r,z)al(r,z) is represented as

|al(r,z)|2s,at=(12π)202π02πum(r,φ,z)um*(r,φ,z)sexp[il(φφ)]×exp[iψ(r,φ)iψ(r,φ)]atdφdφ,
where s,at denotes average over the ensemble of the source and turbulence, s denotes average over the ensemble of the light source and at denotes average over the ensemble of oceanic turbulence. exp[iψ(r,φ)iψ*(r,φ)]atrepresents the coherence function of the phase, um(r,φ,z)um*(r,φ,z)s can be expressed as
um(r,φ,z)um*(r,φ,z)s=exp[im(φφ)]1[w04(1+zζ2)]m+1u=0mu=0mm!(mu)!u!×m!(mu)!u!HVn2m2u(x1)HVn2u(y1)HVn2m2u(x2)HVn2u(y2),=exp[im(φφ)]Γ(r,φ,φ,z)
where x1=rcos(φ), y1=rsin(φ),x2=rcos(φ), and y2=rsin(φ), and Γ(r,φ,φ,z)is given by

Γ(r,φ,φ,z)=1[w04(1+zζ2)]m+1u=0mu=0mm!(mu)!u!m!(mu)!u!.×HVn2m2u(x1)HVn2u(y1)HVn2m2u(x2)HVn2u(y2)

Substituting Eq. (28) into Eq. (24), Eq. (24) can be rewritten as

p(l/m)=(12π)2002π02πΓ(r,φ,φ,z)exp[i(lm)(φφ)],×exp[iψ(r,φ)iψ*(r,φ)]atrdrdφdφ

Since the aberrations introduced by oceanic turbulence are random variables, the coherence function of the phase can be expressed as

exp[iψ(r,φ)iψ*(r,φ)]at=exp{12[ψ(r,φ)ψ*(r,φ)]2at},
where [ψ(r,φ)ψ*(r,φ)]2at is the phase structure function of turbulent aberrations. For Kolmogorov turbulence phase aberrations, the phase structure function of turbulent aberrations is given by [25,26]
[ψ(r,φ)ψ*(r,φ)]2at=2|(r1r)2|5/3/ρocζ5/3.
where ρocς represents the spatial coherence radius of a spherical wave propagating in the oceanic turbulence.

Make use of the relations |(r1r2)|5/3={(r1r2)2}5/6and (r1r2)2=2r2[1cos(φφ)], the coherence function of the phase exp[iψ(r,φ)iψ*(r,φ)]at is given by

exp[iψ(r,φ)iψ*(r,φ)]at=exp{25/6[1cos(φφ)]5/6r5/3ρocζ5/3},

In Markov approximation, i.e. κ=ζκρ, the spatial coherence radius ρocς can be written as

ρocς=[π2k2z3ζ40κ3ϕ(κ,ζ)dκ]3/5.

After a long but straightforward calculation [4], we can obtain the lateral coherence length of the spherical wave in anisotropic oceanic turbulence. The Eq. (34) is reduced to

ρocς=ζ6/5|ϖ|6/5[1.802×107k2z(εη)1/3χT(0.483ϖ20.835ϖ+3.380)]3/5,

Based on the Eq. (30) and Eq. (33), we can obtain the received probability [16]

P(l/m)=p(l/m)/h=p(h/m),
where

p(l/m)=(12π)2002π02πΓ(r,φ,φ,z)exp[i(lm)(φφ)],×exp{25/6[1cos(φφ)]5/6r5/3/ρocς5/3}rdrdφdφ

For l=m, P(l|m)=P(m) is the received probability of transmitting OAM modes m. In addition, when lm, P(l|m)=P(l) is the received probability of transmitting OAM modes with index m followed by detection of OAM mode l.

5. Numerical analysis

We now investigate the statistical properties of OAM modes of the HG vortex beam propagating through oceanic turbulence by numerical simulation. In the following analysis, the calculation parameters of HG vortex beams are λ=0.4173μm, η=1mm, w0=1cm, ϖ=0.5, ε=106, z=150m, χT=108, ζ=2, m=2, l=2, n=2, unless other variable parameters are specified in calculation.

Figure 2 displays the received probability of signal OAM modes of the HG vortex beam propagation through oceanic turbulence for several values of parameters n and m. It can be seen that the received probability of signal OAM modes decreases monotonously with increasing propagation distance. Figure 2(a) indicates that the received probability of signal OAM modes of the LG20 beam is smaller than the HG vortex beam, when the transmission distance is greater. It means that the HG vortex beam has a stronger resistance to the oceanic turbulence than the LG20 beam, and the resistance increases with the increase of transmission distance. In addition, from Fig. 2(a-b), the received probability of signal OAM modes decreases with increasing OAM mode number m for a given propagation distance. Therefore, the HG vortex beam with smaller m is less affected by oceanic turbulence than the LGm0 beam from the aspect of the signal OAM probability in long distance transmission.

 

Fig. 2 Probability of signal OAM modes of the HG vortex beam propagation through oceanic turbulence for several values of parameter n (a) m=2, (b) m=4 respectively.

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The discussion of Fig. 2 is related to the received probability of signal OAM modes. The intermodel crosstalk between channels can be caused by optical OAM beams propagation through oceanic turbulence, which represents the OAM energy level difference Δl=|lm|0. Figure 3 displays the difference between the HG vortex beams and the conventional LG beam propagation through oceanic turbulence causing the received probability of crosstalk OAM modes. We find that the received probability of crosstalk OAM modes of the conventional LG beam is obviously larger than that of the HG vortex beam. That is to say, the effects of oceanic turbulence on the HG vortex beam compared to the conventional LG beam are significantly less. The results will be very useful for underwater optical communications.

 

Fig. 3 Probability of crosstalk OAM modes of the HG vortex beam propagation through oceanic turbulence versus propagation distance z for different modes Δl.

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For the HG vortex beam propagation in oceanic turbulence, the received probability of signal OAM modes will be improved in anisotropic turbulence in Fig. 4. Figure 4 also shows that the received probability of signal OAM modes of the HG vortex beam as functions of propagation distance z for different anisotropic factor ζ. For a given value of propagation distance z, when the anisotropic factor ζ increases, the signal probability of HG vortex beams increases.

 

Fig. 4 Probability of signal OAM modes of the HG vortex beam propagation through oceanic turbulence versus propagation distance z for different anisotropic factor ζ.

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To comprehend the regularity of the received probability of signal OAM modes of the HG vortex beam, we calculate the influence of wavelength λ for different inner scale η in Fig. 5. We deduce from Fig. 5 that, for any given inner scale η, there is an increase of the received probability of signal OAM modes with increasing λ. The result shows that the HG vortex beam with longer wavelength is less affected by oceanic turbulence. In addition, Fig. 5 also shows that the received probability of signal OAM modes increases with the increase of η for any given λ. The phenomenon can be explained that the larger inner scale of turbulent eddy results in small light scattering, and then the small light scattering produces a small energy loss. Thus, the received probability of signal OAM modes of the HG vortex beam can be improved under the greater inner scale of oceanic turbulence.

 

Fig. 5 Probability of signal OAM modes of the HG vortex beam propagation through oceanic turbulence versus wavelength λ for different inner scale η.

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We also analyze the influence of the dissipation rate of temperature variance χt and the source’s transverse size w0 on the received probability of signal OAM modes for the HG vortex beam. From Fig. 6, we can see that, as the source’s transverse size w0 increases, the received probability of signal OAM modes of the HG vortex beam increases. This is caused by the larger source’s transverse size to make a smaller beam spread and beam wander which are caused by the diffraction and refringence of the vacuum and oceanic turbulence, respectively. The smaller beam spread and beam wander lead to the smaller transformations of the space distribution of beams propagation in the link. In addition, as would be expected, the received probability of signal OAM modes of the HG vortex beam decreases with increasing dissipation rate of temperature variance χt from Fig. 6, especially in the larger source’s transverse size w0.

 

Fig. 6 Probability of signal OAM modes of the HG vortex beam propagation through oceanic turbulence versus the dissipation rate of temperature variance χtfor different the source’s transverse size w0.

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To further comprehend the influence of oceanic turbulence parameters, in particular the rate of dissipation of kinetic energy per unit mass of fluid ε and the ratio of temperature and salinity contributions to the refractive index spectrum ϖ on the received probability of the signal OAM mode of the HG vortex beam. Figure 7 shows that the received probability of the signal OAM mode is a function of ε for different ϖ. It is evident from Fig. 7 that the larger value of ϖ will obtain the larger received probability of signal OAM modes. This implies that the salinity fluctuations dominated turbulence affects the HG vortex beam propagation more than the temperature fluctuations dominated turbulence. Figure 7 also shows that the larger rate of dissipation of kinetic energy per unit mass of fluid enhances the detection probability of the signal OAM mode of the HG vortex beam. This conclusion comes from the larger rate of dissipation of kinetic energy per unit mass of fluid reducing oceanic turbulence.

 

Fig. 7 Probability of signal OAM modes of the HG vortex beam propagation through oceanic turbulence versus the rate of dissipation of kinetic energy per unit mass of fluid ε for different the ratio of temperature and salinity contributions to the refractive index spectrum ϖ.

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6. Conclusion

We develop a new angular spectrum of the vortex beam based on the angular spectrum of the HG beam and the vortex factor (kxiky)m, which evolves a new HG vortex beam. Then we establish the model of the received probabilities of OAM modes of the HG vortex beam propagation through anisotropic oceanic turbulence. The influence of the HG vortex beam on the received probabilities of signal and crosstalk OAM modes shows that the HG vortex beam is superior to the LG beam propagation in anisotropic oceanic turbulence for long-distance transmission. We also find that the HG vortex beam has robust turbulent resistance under the condition of anisotropic oceanic turbulence, longer λ, η, ϖ, ε, z, w0 or smaller χT. The HG vortex beam can realize the goal for enhancing the link capacity of underwater optical communication.

Funding

Fundamental Research Funds for the Central Universities (Grant No.JUSRP51716A).

References and links

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References

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  1. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
    [Crossref] [PubMed]
  2. C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94(15), 153901 (2005).
    [Crossref] [PubMed]
  3. Y. Baykal, “Intensity fluctuations of multimode laser beams in underwater medium,” J. Opt. Soc. Am. A 32(4), 593–598 (2015).
    [Crossref] [PubMed]
  4. Y. Wu, Y. Zhang, Y. Li, and Z. Hu, “Beam wander of Gaussian-Schell model beams propagating through oceanic turbulence,” Opt. Commun. 371, 59–66 (2016).
    [Crossref]
  5. V. V. Nikishov and V. I. Nikishov, “Spectrum of turbulent fluctuations of the sea-water refraction index,” Int. J. Fluid Mech. Res. 27(1), 82–98 (2000).
    [Crossref]
  6. M. Cheng, L. Guo, J. Li, Q. Huang, Q. Cheng, and D. Zhang, “Propagation of an optical vortex carried by a partially coherent Laguerre-Gaussian beam in turbulent ocean,” Appl. Opt. 55(17), 4642–4648 (2016).
    [Crossref] [PubMed]
  7. M. Cheng, L. Guo, J. Li, and Y. Zhang, “Channel capacity of the OAM based free-space optical communication links with Bessel–Gauss beams in turbulent ocean,” IEEE Photonics J. 8, 7901411 (2016).
  8. J. Xu and D. Zhao, “Propagation of a stochastic electromagnetic vortex beam in the oceanic turbulence,” Opt. Laser Technol. 57, 189–193 (2014).
    [Crossref]
  9. Y. Huang, B. Zhang, Z. Gao, G. Zhao, and Z. Duan, “Evolution behavior of Gaussian Schell-model vortex beams propagating through oceanic turbulence,” Opt. Express 22(15), 17723–17734 (2014).
    [Crossref] [PubMed]
  10. D. Liu, Y. Wang, and H. Yin, “Evolution properties of partially coherent flat-topped vortex hollow beam in oceanic turbulence,” Appl. Opt. 54(35), 10510–10516 (2015).
    [Crossref] [PubMed]
  11. B. Galperin, S. Sukoriansky, N. Dikovskaya, P. L. Read, Y. H. Yamazaki, and R. Wordsworth, “Anisotropic turbulence and zonal jets in rotating flows with a β-effect,” Nonlinear Process. Geophys. 13(1), 83–98 (2006).
    [Crossref]
  12. 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]
  13. 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 (2006).
    [Crossref]
  14. V. P. Lukin, “Investigation of some peculiarities in the structure of large scale atmospheric turbulence,” Proc. SPIE 2200, 384–395 (1994).
    [Crossref]
  15. V. P. Lukin, “Investigation of the anisotropy of the atmospheric turbulence spectrum in the low-frequency range,” Proc. SPIE 2471, 347–355 (1995).
    [Crossref]
  16. Y. Zhu, M. Chen, Y. Zhang, and Y. Li, “Propagation of the OAM mode carried by partially coherent modified Bessel-Gaussian beams in an anisotropic non-Kolmogorov marine atmosphere,” J. Opt. Soc. Am. A 33(12), 2277–2283 (2016).
    [Crossref] [PubMed]
  17. 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]
  18. Y. Li, Y. Zhang, Y. Zhu, and M. Chen, “Effects of anisotropic turbulence on average polarizability of Gaussian Schell-model quantized beams through ocean link,” Appl. Opt. 55(19), 5234–5239 (2016).
    [Crossref] [PubMed]
  19. Y. Wu, Y. Zhang, Y. Zhu, and Z. Hu, “Spreading and wandering of Gaussian-Schell model laser beams in an anisotropic turbulent ocean,” Laser Phys. 26(9), 095001 (2016).
    [Crossref]
  20. W. Lu, L. Liu, and J. F. Sun, “Influence of temperature and salinity fluctuations on propagation behaviour of partially coherent beams in oceanic turbulence,” J. Opt. A, Pure Appl. Opt. 8(12), 1052–1058 (2006).
    [Crossref]
  21. F. Pampaloni and J. Enderlein, “Gaussian, Hermite-Gaussian, and Laguerre-Gaussian beams: A primer,” arXiv:physics/0410021 (2004)
  22. O. Korotkova, Random Light Beams Theory and Applications (CRC, 2014).
  23. I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, Series and Products, 6th Ed. (Academic, 2000).
  24. V. A. Sennikov, P. A. Konyaev, and V. P. Lukin, “Computer simulation of scalar vortex and annular beams LG0L in time-varying random inhomogeneous media,” Proc. SPIE 10035, 10035–10237 (2016).
  25. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (Washington: SPIE, 2005).
  26. X. Sheng, Y. Zhang, F. Zhao, L. Zhang, and Y. Zhu, “Effects of low-order atmosphere-turbulence aberrations on the entangled orbital angular momentum states,” Opt. Lett. 37(13), 2607–2609 (2012).
    [Crossref] [PubMed]

2016 (7)

Y. Wu, Y. Zhang, Y. Li, and Z. Hu, “Beam wander of Gaussian-Schell model beams propagating through oceanic turbulence,” Opt. Commun. 371, 59–66 (2016).
[Crossref]

M. Cheng, L. Guo, J. Li, Q. Huang, Q. Cheng, and D. Zhang, “Propagation of an optical vortex carried by a partially coherent Laguerre-Gaussian beam in turbulent ocean,” Appl. Opt. 55(17), 4642–4648 (2016).
[Crossref] [PubMed]

M. Cheng, L. Guo, J. Li, and Y. Zhang, “Channel capacity of the OAM based free-space optical communication links with Bessel–Gauss beams in turbulent ocean,” IEEE Photonics J. 8, 7901411 (2016).

Y. Zhu, M. Chen, Y. Zhang, and Y. Li, “Propagation of the OAM mode carried by partially coherent modified Bessel-Gaussian beams in an anisotropic non-Kolmogorov marine atmosphere,” J. Opt. Soc. Am. A 33(12), 2277–2283 (2016).
[Crossref] [PubMed]

Y. Li, Y. Zhang, Y. Zhu, and M. Chen, “Effects of anisotropic turbulence on average polarizability of Gaussian Schell-model quantized beams through ocean link,” Appl. Opt. 55(19), 5234–5239 (2016).
[Crossref] [PubMed]

Y. Wu, Y. Zhang, Y. Zhu, and Z. Hu, “Spreading and wandering of Gaussian-Schell model laser beams in an anisotropic turbulent ocean,” Laser Phys. 26(9), 095001 (2016).
[Crossref]

V. A. Sennikov, P. A. Konyaev, and V. P. Lukin, “Computer simulation of scalar vortex and annular beams LG0L in time-varying random inhomogeneous media,” Proc. SPIE 10035, 10035–10237 (2016).

2015 (2)

2014 (3)

2012 (1)

2006 (3)

W. Lu, L. Liu, and J. F. Sun, “Influence of temperature and salinity fluctuations on propagation behaviour of partially coherent beams in oceanic turbulence,” J. Opt. A, Pure Appl. Opt. 8(12), 1052–1058 (2006).
[Crossref]

B. Galperin, S. Sukoriansky, N. Dikovskaya, P. L. Read, Y. H. Yamazaki, and R. Wordsworth, “Anisotropic turbulence and zonal jets in rotating flows with a β-effect,” Nonlinear Process. Geophys. 13(1), 83–98 (2006).
[Crossref]

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

2005 (1)

C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94(15), 153901 (2005).
[Crossref] [PubMed]

2000 (1)

V. V. Nikishov and V. I. Nikishov, “Spectrum of turbulent fluctuations of the sea-water refraction index,” Int. J. Fluid Mech. Res. 27(1), 82–98 (2000).
[Crossref]

1999 (1)

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]

1995 (1)

V. P. Lukin, “Investigation of the anisotropy of the atmospheric turbulence spectrum in the low-frequency range,” Proc. SPIE 2471, 347–355 (1995).
[Crossref]

1994 (1)

V. P. Lukin, “Investigation of some peculiarities in the structure of large scale atmospheric turbulence,” Proc. SPIE 2200, 384–395 (1994).
[Crossref]

1992 (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Allen, L.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

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]

Baykal, Y.

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

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 (2006).
[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]

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]

Chen, M.

Cheng, M.

M. Cheng, L. Guo, J. Li, Q. Huang, Q. Cheng, and D. Zhang, “Propagation of an optical vortex carried by a partially coherent Laguerre-Gaussian beam in turbulent ocean,” Appl. Opt. 55(17), 4642–4648 (2016).
[Crossref] [PubMed]

M. Cheng, L. Guo, J. Li, and Y. Zhang, “Channel capacity of the OAM based free-space optical communication links with Bessel–Gauss beams in turbulent ocean,” IEEE Photonics J. 8, 7901411 (2016).

Cheng, Q.

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 (2006).
[Crossref]

Dikovskaya, N.

B. Galperin, S. Sukoriansky, N. Dikovskaya, P. L. Read, Y. H. Yamazaki, and R. Wordsworth, “Anisotropic turbulence and zonal jets in rotating flows with a β-effect,” Nonlinear Process. Geophys. 13(1), 83–98 (2006).
[Crossref]

Duan, Z.

Fugate, R. Q.

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]

Galperin, B.

B. Galperin, S. Sukoriansky, N. Dikovskaya, P. L. Read, Y. H. Yamazaki, and R. Wordsworth, “Anisotropic turbulence and zonal jets in rotating flows with a β-effect,” Nonlinear Process. Geophys. 13(1), 83–98 (2006).
[Crossref]

Gao, Z.

Guo, L.

M. Cheng, L. Guo, J. Li, and Y. Zhang, “Channel capacity of the OAM based free-space optical communication links with Bessel–Gauss beams in turbulent ocean,” IEEE Photonics J. 8, 7901411 (2016).

M. Cheng, L. Guo, J. Li, Q. Huang, Q. Cheng, and D. Zhang, “Propagation of an optical vortex carried by a partially coherent Laguerre-Gaussian beam in turbulent ocean,” Appl. Opt. 55(17), 4642–4648 (2016).
[Crossref] [PubMed]

Hu, Z.

Y. Wu, Y. Zhang, Y. Li, and Z. Hu, “Beam wander of Gaussian-Schell model beams propagating through oceanic turbulence,” Opt. Commun. 371, 59–66 (2016).
[Crossref]

Y. Wu, Y. Zhang, Y. Zhu, and Z. Hu, “Spreading and wandering of Gaussian-Schell model laser beams in an anisotropic turbulent ocean,” Laser Phys. 26(9), 095001 (2016).
[Crossref]

Huang, Q.

Huang, Y.

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 (2006).
[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]

Konyaev, P. A.

V. A. Sennikov, P. A. Konyaev, and V. P. Lukin, “Computer simulation of scalar vortex and annular beams LG0L in time-varying random inhomogeneous media,” Proc. SPIE 10035, 10035–10237 (2016).

Li, J.

M. Cheng, L. Guo, J. Li, and Y. Zhang, “Channel capacity of the OAM based free-space optical communication links with Bessel–Gauss beams in turbulent ocean,” IEEE Photonics J. 8, 7901411 (2016).

M. Cheng, L. Guo, J. Li, Q. Huang, Q. Cheng, and D. Zhang, “Propagation of an optical vortex carried by a partially coherent Laguerre-Gaussian beam in turbulent ocean,” Appl. Opt. 55(17), 4642–4648 (2016).
[Crossref] [PubMed]

Li, Y.

Liu, D.

Liu, L.

W. Lu, L. Liu, and J. F. Sun, “Influence of temperature and salinity fluctuations on propagation behaviour of partially coherent beams in oceanic turbulence,” J. Opt. A, Pure Appl. Opt. 8(12), 1052–1058 (2006).
[Crossref]

Lu, W.

W. Lu, L. Liu, and J. F. Sun, “Influence of temperature and salinity fluctuations on propagation behaviour of partially coherent beams in oceanic turbulence,” J. Opt. A, Pure Appl. Opt. 8(12), 1052–1058 (2006).
[Crossref]

Lukin, V. P.

V. A. Sennikov, P. A. Konyaev, and V. P. Lukin, “Computer simulation of scalar vortex and annular beams LG0L in time-varying random inhomogeneous media,” Proc. SPIE 10035, 10035–10237 (2016).

V. P. Lukin, “Investigation of the anisotropy of the atmospheric turbulence spectrum in the low-frequency range,” Proc. SPIE 2471, 347–355 (1995).
[Crossref]

V. P. Lukin, “Investigation of some peculiarities in the structure of large scale atmospheric turbulence,” Proc. SPIE 2200, 384–395 (1994).
[Crossref]

Nikishov, V. I.

V. V. Nikishov and V. I. Nikishov, “Spectrum of turbulent fluctuations of the sea-water refraction index,” Int. J. Fluid Mech. Res. 27(1), 82–98 (2000).
[Crossref]

Nikishov, V. V.

V. V. Nikishov and V. I. Nikishov, “Spectrum of turbulent fluctuations of the sea-water refraction index,” Int. J. Fluid Mech. Res. 27(1), 82–98 (2000).
[Crossref]

Osmon, C. L.

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]

Paterson, C.

C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94(15), 153901 (2005).
[Crossref] [PubMed]

Read, P. L.

B. Galperin, S. Sukoriansky, N. Dikovskaya, P. L. Read, Y. H. Yamazaki, and R. Wordsworth, “Anisotropic turbulence and zonal jets in rotating flows with a β-effect,” Nonlinear Process. Geophys. 13(1), 83–98 (2006).
[Crossref]

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 (2006).
[Crossref]

Sennikov, V. A.

V. A. Sennikov, P. A. Konyaev, and V. P. Lukin, “Computer simulation of scalar vortex and annular beams LG0L in time-varying random inhomogeneous media,” Proc. SPIE 10035, 10035–10237 (2016).

Sheng, X.

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Sukoriansky, S.

B. Galperin, S. Sukoriansky, N. Dikovskaya, P. L. Read, Y. H. Yamazaki, and R. Wordsworth, “Anisotropic turbulence and zonal jets in rotating flows with a β-effect,” Nonlinear Process. Geophys. 13(1), 83–98 (2006).
[Crossref]

Sun, J. F.

W. Lu, L. Liu, and J. F. Sun, “Influence of temperature and salinity fluctuations on propagation behaviour of partially coherent beams in oceanic turbulence,” J. Opt. A, Pure Appl. Opt. 8(12), 1052–1058 (2006).
[Crossref]

Toselli, I.

Wang, Y.

Woerdman, J. P.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Wordsworth, R.

B. Galperin, S. Sukoriansky, N. Dikovskaya, P. L. Read, Y. H. Yamazaki, and R. Wordsworth, “Anisotropic turbulence and zonal jets in rotating flows with a β-effect,” Nonlinear Process. Geophys. 13(1), 83–98 (2006).
[Crossref]

Wu, Y.

Y. Wu, Y. Zhang, Y. Zhu, and Z. Hu, “Spreading and wandering of Gaussian-Schell model laser beams in an anisotropic turbulent ocean,” Laser Phys. 26(9), 095001 (2016).
[Crossref]

Y. Wu, Y. Zhang, Y. Li, and Z. Hu, “Beam wander of Gaussian-Schell model beams propagating through oceanic turbulence,” Opt. Commun. 371, 59–66 (2016).
[Crossref]

Xu, J.

J. Xu and D. Zhao, “Propagation of a stochastic electromagnetic vortex beam in the oceanic turbulence,” Opt. Laser Technol. 57, 189–193 (2014).
[Crossref]

Yamazaki, Y. H.

B. Galperin, S. Sukoriansky, N. Dikovskaya, P. L. Read, Y. H. Yamazaki, and R. Wordsworth, “Anisotropic turbulence and zonal jets in rotating flows with a β-effect,” Nonlinear Process. Geophys. 13(1), 83–98 (2006).
[Crossref]

Yin, H.

Zhang, B.

Zhang, D.

Zhang, L.

Zhang, Y.

Y. Wu, Y. Zhang, Y. Li, and Z. Hu, “Beam wander of Gaussian-Schell model beams propagating through oceanic turbulence,” Opt. Commun. 371, 59–66 (2016).
[Crossref]

M. Cheng, L. Guo, J. Li, and Y. Zhang, “Channel capacity of the OAM based free-space optical communication links with Bessel–Gauss beams in turbulent ocean,” IEEE Photonics J. 8, 7901411 (2016).

Y. Zhu, M. Chen, Y. Zhang, and Y. Li, “Propagation of the OAM mode carried by partially coherent modified Bessel-Gaussian beams in an anisotropic non-Kolmogorov marine atmosphere,” J. Opt. Soc. Am. A 33(12), 2277–2283 (2016).
[Crossref] [PubMed]

Y. Wu, Y. Zhang, Y. Zhu, and Z. Hu, “Spreading and wandering of Gaussian-Schell model laser beams in an anisotropic turbulent ocean,” Laser Phys. 26(9), 095001 (2016).
[Crossref]

Y. Li, Y. Zhang, Y. Zhu, and M. Chen, “Effects of anisotropic turbulence on average polarizability of Gaussian Schell-model quantized beams through ocean link,” Appl. Opt. 55(19), 5234–5239 (2016).
[Crossref] [PubMed]

X. Sheng, Y. Zhang, F. Zhao, L. Zhang, and Y. Zhu, “Effects of low-order atmosphere-turbulence aberrations on the entangled orbital angular momentum states,” Opt. Lett. 37(13), 2607–2609 (2012).
[Crossref] [PubMed]

Zhao, D.

J. Xu and D. Zhao, “Propagation of a stochastic electromagnetic vortex beam in the oceanic turbulence,” Opt. Laser Technol. 57, 189–193 (2014).
[Crossref]

Zhao, F.

Zhao, G.

Zhu, Y.

Appl. Opt. (3)

IEEE Photonics J. (1)

M. Cheng, L. Guo, J. Li, and Y. Zhang, “Channel capacity of the OAM based free-space optical communication links with Bessel–Gauss beams in turbulent ocean,” IEEE Photonics J. 8, 7901411 (2016).

Int. J. Fluid Mech. Res. (1)

V. V. Nikishov and V. I. Nikishov, “Spectrum of turbulent fluctuations of the sea-water refraction index,” Int. J. Fluid Mech. Res. 27(1), 82–98 (2000).
[Crossref]

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

W. Lu, L. Liu, and J. F. Sun, “Influence of temperature and salinity fluctuations on propagation behaviour of partially coherent beams in oceanic turbulence,” J. Opt. A, Pure Appl. Opt. 8(12), 1052–1058 (2006).
[Crossref]

J. Opt. Soc. Am. A (3)

Laser Phys. (1)

Y. Wu, Y. Zhang, Y. Zhu, and Z. Hu, “Spreading and wandering of Gaussian-Schell model laser beams in an anisotropic turbulent ocean,” Laser Phys. 26(9), 095001 (2016).
[Crossref]

Nonlinear Process. Geophys. (1)

B. Galperin, S. Sukoriansky, N. Dikovskaya, P. L. Read, Y. H. Yamazaki, and R. Wordsworth, “Anisotropic turbulence and zonal jets in rotating flows with a β-effect,” Nonlinear Process. Geophys. 13(1), 83–98 (2006).
[Crossref]

Opt. Commun. (1)

Y. Wu, Y. Zhang, Y. Li, and Z. Hu, “Beam wander of Gaussian-Schell model beams propagating through oceanic turbulence,” Opt. Commun. 371, 59–66 (2016).
[Crossref]

Opt. Express (1)

Opt. Laser Technol. (1)

J. Xu and D. Zhao, “Propagation of a stochastic electromagnetic vortex beam in the oceanic turbulence,” Opt. Laser Technol. 57, 189–193 (2014).
[Crossref]

Opt. Lett. (1)

Phys. Rev. A (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Phys. Rev. Lett. (1)

C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94(15), 153901 (2005).
[Crossref] [PubMed]

Proc. SPIE (5)

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, 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 (2006).
[Crossref]

V. P. Lukin, “Investigation of some peculiarities in the structure of large scale atmospheric turbulence,” Proc. SPIE 2200, 384–395 (1994).
[Crossref]

V. P. Lukin, “Investigation of the anisotropy of the atmospheric turbulence spectrum in the low-frequency range,” Proc. SPIE 2471, 347–355 (1995).
[Crossref]

V. A. Sennikov, P. A. Konyaev, and V. P. Lukin, “Computer simulation of scalar vortex and annular beams LG0L in time-varying random inhomogeneous media,” Proc. SPIE 10035, 10035–10237 (2016).

Other (4)

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (Washington: SPIE, 2005).

F. Pampaloni and J. Enderlein, “Gaussian, Hermite-Gaussian, and Laguerre-Gaussian beams: A primer,” arXiv:physics/0410021 (2004)

O. Korotkova, Random Light Beams Theory and Applications (CRC, 2014).

I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, Series and Products, 6th Ed. (Academic, 2000).

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Figures (7)

Fig. 1
Fig. 1 Intensity profiles of HG vortex beams in the absence of turbulence (a) n = 2 , m = 2 , (b) n = 2 , m = 4 ,(c) n = 2 , m = 6 . When z = 0 in the plane, the intensity of the HG vortex beam decreases with deviating from the optical axis center. In addition, the intensity forms annular shapes along the optical axis center with increasing quantum number m .
Fig. 2
Fig. 2 Probability of signal OAM modes of the HG vortex beam propagation through oceanic turbulence for several values of parameter n (a) m = 2 , (b) m = 4 respectively.
Fig. 3
Fig. 3 Probability of crosstalk OAM modes of the HG vortex beam propagation through oceanic turbulence versus propagation distance z for different modes Δ l .
Fig. 4
Fig. 4 Probability of signal OAM modes of the HG vortex beam propagation through oceanic turbulence versus propagation distance z for different anisotropic factor ζ .
Fig. 5
Fig. 5 Probability of signal OAM modes of the HG vortex beam propagation through oceanic turbulence versus wavelength λ for different inner scale η .
Fig. 6
Fig. 6 Probability of signal OAM modes of the HG vortex beam propagation through oceanic turbulence versus the dissipation rate of temperature variance χ t for different the source’s transverse size w 0 .
Fig. 7
Fig. 7 Probability of signal OAM modes of the HG vortex beam propagation through oceanic turbulence versus the rate of dissipation of kinetic energy per unit mass of fluid ε for different the ratio of temperature and salinity contributions to the refractive index spectrum ϖ .

Equations (37)

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ϕ ( κ , ζ ) = 0.388 C m 2 ζ 2 κ 11 / 3 [ 1 + 2.35 ( κ η ) 2 / 3 ] f ( κ , ζ , ϖ ) ,
u ( x , y , z ; ω ) = exp ( i k z ) 4 π 2 a ( k x , k y ; ω ) exp [ i ( k x x + k y y ) ] exp [ i z 2 k ( k x 2 + k y 2 ) ] d k x d k y ,
Δ θ ( z ; ω ) Δ ω ( z ; ω ) min .
Δ θ ( z ; ω ) = 1 4 π 2 ( k x 2 + k y 2 ) | a ( k x , k y ; ω ) | 2 d k x d k y = 1 4 π 2 k x 2 | a ( k x , k y ; ω ) | 2 d k x k y 2 | a ( k x , k y ; ω ) | 2 d k y ,
Δ ω ( z ; ω ) = 1 4 π 2 ( | a ( k x , k y ; ω ) k x | 2 + | a ( k x , k y ; ω ) k y | 2 ) d k x d k y , = 1 4 π 2 | a ( k x , k y ; ω ) k x | 2 d k x | a ( k x , k y ; ω ) k y | 2 d k y
1 4 π 2 k x 2 | a ( k x , k y ; ω ) | 2 d k x | a ( k x , k y ; ω ) k x | 2 d k min .
| f ( x ) | 2 d x | g ( x ) | 2 d x | f * ( x ) g ( x ) d x | 2 ,
a ( k x , k y ; ω ) = exp [ w 0 2 4 ( k x 2 + k y 2 ) ] .
a ( k x , k y ; ω ) = k x n k y n exp [ w 0 2 4 ( k x 2 + k y 2 ) ] .
a ( k x , k y ; ω ) = k x n k y n ( k x i k y ) m exp [ w 0 2 4 ( k x 2 + k y 2 ) ] ,
u ( x , y , z ; ω ) = exp ( i k z ) 4 π 2 k x n k y n ( k x i k y ) m exp [ ϒ ( k x , k y ) ] d k x d k y ,
exp [ ϒ ( k x , k y ) ] = exp [ w 0 2 4 ( 1 + i z ζ ) ( k x 2 + k y 2 ) ] exp [ i ( k x x + k y y ) ] ,
n exp [ i ( k x x + k y y ) ] x n = ( i k x ) n exp [ i ( k x x + k y y ) ] ,
n exp [ i ( k x x + k y y ) ] y n = ( i k y ) n exp [ i ( k x x + k y y ) ] ,
( x i y ) m exp [ i ( k x x + k y y ) ] = i m ( k x i k y ) m exp [ i ( k x x + k y y ) ] .
u ( x , y , z ; ω ) = exp ( i k z ) 4 π 2 2 n x n y n ( x i y ) m exp [ ϒ ( k x , k y ) ] d k x d k y .
u ( x , y , z ; ω ) = exp ( i k z ) 4 π 2 2 n x n y n γ a m exp [ ϒ ( k x , k y ) ] d k x d k y ,
1 2 π exp ( i a x b 2 2 x 2 ) d x = 1 b 2 π exp [ a 2 2 b 2 ] ,
u m ( x , y , z ; ω ) = exp ( i k z ) 1 [ w 0 2 ( 1 + i z ζ ) ] m + 1 2 n x n y n γ b m exp [ γ a γ b w 0 2 ( 1 + i z ζ ) ] .
u m ( x , y , z ; ω ) = exp ( i m φ ) exp ( i k z ) [ w 0 2 ( 1 + i z ζ ) ] m + 1 u = 0 m / 2 ( m / 2 ) ! ( m / 2 u ) ! u ! H V n m 2 u ( x ) H V n 2 u ( y ) ,
u m ( x , y , z ; ω ) = 1 [ w 0 2 ( 1 + i z ζ ) ] n + 1 H n ( x / w 0 2 ( 1 + i z ζ ) ) × H n ( y w 0 2 ( 1 + i z ζ ) ) exp [ i k z x 2 + y 2 w 0 2 ( 1 + i z ζ ) ] ,
u m ( x , y , z ; ω ) = exp ( i m φ ) exp ( i k z ) [ w 0 2 ( 1 + i z ζ ) ] m + 1 ( r ) m exp [ r 2 w 0 2 ( 1 + i z ζ ) ] L m 0 [ r 2 w 0 2 ( 1 + i z ζ ) ] ,
p ( l | u ) = l | a l ( r , z ) | 2 ,
p ( l / m ) = p ( l | u ) = m | a l ( r , z ) | 2 ,
a l ( r , z ) = 1 2 π 0 2 π u ( r , φ , z ) exp ( i l φ ) d φ .
u ( r , φ , z ) = u m ( r , φ , z ) exp [ i ψ ( r , φ ) ] ,
| a l ( r , z ) | 2 s , a t = ( 1 2 π ) 2 0 2 π 0 2 π u m ( r , φ , z ) u m * ( r , φ , z ) s exp [ i l ( φ φ ) ] × exp [ i ψ ( r , φ ) i ψ ( r , φ ) ] a t d φ d φ ,
u m ( r , φ , z ) u m * ( r , φ , z ) s = exp [ i m ( φ φ ) ] 1 [ w 0 4 ( 1 + z ζ 2 ) ] m + 1 u = 0 m u = 0 m m ! ( m u ) ! u ! × m ! ( m u ) ! u ! H V n 2 m 2 u ( x 1 ) H V n 2 u ( y 1 ) H V n 2 m 2 u ( x 2 ) H V n 2 u ( y 2 ) , = exp [ i m ( φ φ ) ] Γ ( r , φ , φ , z )
Γ ( r , φ , φ , z ) = 1 [ w 0 4 ( 1 + z ζ 2 ) ] m + 1 u = 0 m u = 0 m m ! ( m u ) ! u ! m ! ( m u ) ! u ! . × H V n 2 m 2 u ( x 1 ) H V n 2 u ( y 1 ) H V n 2 m 2 u ( x 2 ) H V n 2 u ( y 2 )
p ( l / m ) = ( 1 2 π ) 2 0 0 2 π 0 2 π Γ ( r , φ , φ , z ) exp [ i ( l m ) ( φ φ ) ] , × exp [ i ψ ( r , φ ) i ψ * ( r , φ ) ] a t r d r d φ d φ
exp [ i ψ ( r , φ ) i ψ * ( r , φ ) ] a t = exp { 1 2 [ ψ ( r , φ ) ψ * ( r , φ ) ] 2 a t } ,
[ ψ ( r , φ ) ψ * ( r , φ ) ] 2 a t = 2 | ( r 1 r ) 2 | 5 / 3 / ρ o c ζ 5 / 3 .
exp [ i ψ ( r , φ ) i ψ * ( r , φ ) ] a t = exp { 2 5 / 6 [ 1 cos ( φ φ ) ] 5 / 6 r 5 / 3 ρ o c ζ 5 / 3 } ,
ρ o c ς = [ π 2 k 2 z 3 ζ 4 0 κ 3 ϕ ( κ , ζ ) d κ ] 3 / 5 .
ρ o c ς = ζ 6 / 5 | ϖ | 6 / 5 [ 1.802 × 10 7 k 2 z ( ε η ) 1 / 3 χ T ( 0.483 ϖ 2 0.835 ϖ + 3.380 ) ] 3 / 5 ,
P ( l / m ) = p ( l / m ) / h = p ( h / m ) ,
p ( l / m ) = ( 1 2 π ) 2 0 0 2 π 0 2 π Γ ( r , φ , φ , z ) exp [ i ( l m ) ( φ φ ) ] , × exp { 2 5 / 6 [ 1 cos ( φ φ ) ] 5 / 6 r 5 / 3 / ρ o c ς 5 / 3 } r d r d φ d φ

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