Abstract

The spatial coherence radius in moderate-to-strong maritime turbulence is derived on the basis of the modified Rytov approximation. Models are developed to simulate the spiral spectrum of Airy beams propagating through moderate-to-strong maritime turbulence. In the moderate-to-strong irradiance fluctuation region, we analyze the effects of maritime turbulence on the spread of the spiral spectrum of Airy beams in a horizontal propagation path. Results indicate that the increment in the inner-scale significantly increases the received power. By contrast, the outer-scale elicits a negligible effect on the received power if the ratio of the inner-scale to the outer-scale is less than 0.01. The outer-scale affects the received power only if the ratio is greater than 0.01. The performance of a light source is essential for the received power of Airy beams carrying orbital angular momentum (OAM) through moderate-to-strong maritime turbulence. Airy beams with longer wavelengths, smaller OAM numbers, larger radii of the main ring, and smaller diameters of the circular aperture are less affected by maritime turbulence. Autofocusing of Airy beams is beneficial for the propagation of the spiral spectrum in a certain propagation distance. These results contribute to the design of optical communication systems with OAM encoding for moderate-to-strong maritime turbulence.

© 2016 Optical Society of America

1. Introduction

Understanding the characteristics of optical propagation through atmospheric turbulence in the maritime environment is attractive and complementary for free space optical communications between marine vessels. Humidity and temperature fluctuations in maritime environments may complicate optical propagation to a greater extent than those in terrestrial environments; the correlation and cospectrum of temperature and humidity fluctuations must be considered [1]. According to Hill’s experimental data [2], a new marine atmospheric spectrum that describes atmospheric turbulence in a general maritime environment is proposed [3]. Considering the effect of the inner-scale of turbulence, researchers [3] developed the irradiance fluctuation expressions of plane and spherical waves on the basis of the new maritime spectrum in weak optical turbulence. In maritime environments, Vetelino et al. [4] proposed theoretical scintillation expressions under moderate-to-strong Kolmogorov turbulence. In weak non-Kolmogorov maritime atmospheric turbulence channels, scintillation and aperture averaging of Gaussian beams were investigated [5].

The propagation of a spiral spectrum with orbital angular momentum (OAM) through atmospheric turbulence has been extensively investigated in the field of optical wireless communications owing to its potential use in multivalued encoding of information and applications in communication systems [6–10]. A key motivation for this idea is that OAM can take any integer value and span an infinite dimensional orthonormal basis [11, 12]. This property may provide a potential solution in the context of wireless communications [12, 13]. However, OAM is susceptible to atmospheric turbulence because of its natural spatial structure. Atmospheric turbulence can induce the spread of the spiral spectrum of OAM, crosstalk among neighboring channels, and attenuation of the information capacity of communication channels. Several models and experiments have recently been established and implemented for the spread of the input mode power over neighboring OAM modes, or a spiral spectrum, resulting in crosstalk between the channels through weak fluctuation region [8, 14–16]. To mitigate the effect of turbulence on OAM, non-diffracting beams [8, 9, 15] are adopted to reduce scattering artefacts and to increase the quality of OAM encoding. Airy beam, a member of the non-diffracting family, is a potential candidate because of several properties, including non-diffraction [17], self-healing [18, 19], and autofocusing [20, 21].

In this study, the spiral spectrum of Airy beams is investigated in the moderate-to-strong turbulence of maritime atmosphere. We initially develop the theoretical expression of the spatial coherence radius through moderate-to-strong maritime turbulence via the modified Rytov approximation. We then obtain the expressions of the probability distribution function and normalized received power of OAM as Airy beams propagate in moderate-to-strong maritime turbulence.

2. Effective Kolmogorov spectrum for moderate-to-strong maritime turbulence

Kolmogorov spectrum is the traditional spatial power spectrum of refractive-index fluctuations when laser beams propagate in weak atmospheric turbulence. This spectrum is described by the following expression

ϕn(κ)=0.033Cn2κ11/3,0κ<,
where κ is the spatial wave number and Cn2 is the refractive-index structure parameter.

Rytov approximation is extended to the moderate-to-strong fluctuation regime because of theoretically investigation of the inner-scale effects under weak fluctuations and outer-scale effects under strong fluctuations. To develop a model under moderate-to-strong turbulence, spatial frequency filter functions of the refractive index spectrum are introduced and the Kolmogorov spectrum is replaced by the effective Kolmogorov spectrum [22]

ϕn,eff(κ)=ϕn(κ)[f(κl0)g(κL0)Gx(κ)+Gy(κ)]=ϕn(κ)[Gx(κ,l0,L0)+Gy(κ)],
where f(κl0) and g(κL0) are the factors describing the inner- and outer-scale effects, respectively. l0 is the inner-scale of turbulence, typically in the order of millimeters. L0 is defined as the outer-scale of turbulence, typically in the order of meters. If the inner- and outer-scale effects are ignored, then f(κl0)=1,g(κL0)=1. Gx(κ) and Gy(κ) are the large-scale and small-scale filter functions, respectively. Meanwhile, Gx(κ,l0,L0)=f(κl0)g(κL0)Gx(κ).

If the inner-scale effects are important, the expression of f(κl0) for a maritime environment is defined by [3, 4]

f(κl0)=exp(κ2κH2)[10.061κκH+2.836(κκH)7/6],κH=3.41l0,
where κH is the spatial wave number associated with the inner-scale of maritime atmospheric turbulence.

The outer-scale effects are adopted by denoting g(κL0) as [22]

g(κL0)=1exp(κ2κ02),
where κ0=8π/L0 is the spatial wave number related to the outer-scale of maritime atmospheric turbulence.

The filter functions Gx(κ) and Gy(κ) are identified as [22]

Gx(κ)=exp(κ2κx2),
Gy(κ)=κ11/3(κ2+κy2)11/6,
where the quantities κx and κy denote the spatial-frequency cutoff for the large-scale and small-scale turbulent cell effects, respectively. κx2=kηx/z, where k is the wave number and zis the propagation distance. In the case of a finite aperture receiver and a finite outer-scale, the quantity ηx of a plane wave is defined by [3, 4, 22]
ηx2.61Q0/[2.61+Q0(1+0.65d2+0.45σR2QH1/6)],
where Q0=zκ02/k. d=kD2/(4z) is the circular aperture radius D/2 scaled by the Fresnel zone z/k. Meanwhile, QH=11.628z/(kl02) and σR2=1.23Cn2k7/6z11/6 is the Rytov variance. κy2=kηy/z, the parameter ηy is introduced as follows [22]
ηy=3(σRσP)12/5(1+0.69σP12/5),
where σP is the plane wave scintillation for the marine spectrum and given by [3]

σP2=3.86σ12{(1+1QH2)11/12[sin(116tan1QH)0.051sin(43tan1QH)(1+QH2)1/4+3.052sin(54tan1QH)(1+QH2)7/24]5.581QH5/6}.

At this point, the effective marine spectrum for a receiver with a finite aperture in moderate-to-strong maritime atmospheric turbulence becomes

ϕn,eff(κ)=0.033Cn2{1[κ2+κy2]11/6+κ11/3[10.061κκH+2.836(κκH)7/6][exp(κ2κxH2)exp(κ2κx0H2)]},
where κxH2=κx2κH2κx2+κH2,κx0H2=κH2κx02κH2+κx02andκx02=κx2κ02κx2+κ02.

3. Probability distribution and normalized received power of spiral spectrum mode

Consider an aperture truncated Airy beam that initially has a transverse spatial wave function corresponding to an eigenstate of OAM. We write the initial radial Airy profile at the source plane (z = 0) in cylindrical coordinates as [21, 23]

Ai0(r,φ,z=0)=Ai[±(r0rω0)]exp[±a(r0rω0)]exp(im0φ),
where r=|r|,r=(x,y) is the two-dimensional position vector in the source plane; φ is the azimuthal angle; r0 is approximately the radius of the main ring; a is the exponential truncation parameter; ω0 is associated with the arbitrary transverse scale; m0 corresponds to the orbital angular momentum m0 that carried by the beam and describes the helical structure of the wave front around a wave front singularity.

Under the paraxial appromation, the normalized Airy model Ai0(r,φ,z) in the z plane through free-space assumes the following form [21]

Ai0(r,φ,z)=ikzω0(r0ω0a2)Jm0(krr0z)exp(ikr22z+a33im0φ),
where Jm0(krr0/z) denotes the m0th order Bessel function.

In the regions of moderate-to-strong maritime atmospheric turbulence and in the half-space z>0, the complex amplitude of the Airy beam can be written as

Ai(r,φ,z)=Ai0(r,φ,z)exp[ψx(r,φ,z)+ψy(r,φ,z)],
where ψx(r,φ,z) and ψy(r,φ,z) are the complex phase perturbations due to large-scale and small-scale turbulence eddies, respectively.

As beam propagates through the maritime atmospheric turbulence, the effect of the refractive index fluctuations perturbs the complex amplitude of the wave so that it is no longer guaranteed to be in the original eigenstate of orbital angular momentum. The function Ai(r,φ,z) can be written as a superposition of plane waves with phase exp(imφ) [24] as follows

Ai(r,φ,z)=12πmβm(r,z)exp(imφ),
where βm(r,z) is given by the integral

βm(r,z)=12π02πAi(r,φ,z)exp(imφ)dφ.

Given the modified Rytov method [22] and by combining Eqs. (13) and (15), the ensemble averaging of the mode probability density of Airy beams in paraxial channel is given by

|βm(r,z)|2=12π02π02πAi0(r,φ,z)Ai0(r,φ,z)exp[im(φφ)]×exp[ψx(r,φ,z)+ψx(r,φ,z)+ψy(r,φ,z)+ψy(r,φ,z)]dφdφ.

Using the form of the phase structure function given in [25,26], we obtain

exp[ψx(r,φ,z)+ψx(r,φ,z)+ψy(r,φ,z)+ψy(r,φ,z)]=exp{π2k2z3[2r22r2cos(φφ)]0κ3ϕn,eff(κ)dκ}=exp{[2r22r2cos(φφ)](1ρ0x2+1ρ0y2)},
where 0κ3ϕn,eff(κ)dκ=0.033Cn20κ2/3[Gx(κ,l0,L0)+Gy(κ)]dκ and the factors involving the transverse spatial coherence radius are defined as
ρ0x2=π2k2z30.033Cn20κ2/3Gx(κ,l0,L0)dκ=0.054k2Cn2z{Γ(16)[(κxH2)1/6(κx0H2)1/6]0.061κHΓ(23)[(κxH2)2/3(κx0H2)2/3]+2.836κH7/6Γ(34)[(κxH2)3/4(κx0H2)3/4]},ρ0y2=π2k2z30.033Cn20κHκ2/3Gy(κ)dκ=0.027k2zCn2κy11/3κH4F21(116;2;3;κH2κy2).
Meanwhile, we assume
exp[ψx(r,φ,z)+ψx(r,φ,z)+ψy(r,φ,z)+ψy(r,φ,z)]=exp[2r22r2cos(φφ)ρ0xy2],
where ρ0xy is the spatial coherence radius of a plane wave propagating in the moderate-to-strong turbulence, given by

ρ0xy=(0.054Cn2k2z)1/2×{Γ(16)[(κxH2)1/6(κx0H2)1/6]0.061κHΓ(23)[(κxH2)2/3(κx0H2)2/3]+2.836κH7/6Γ(34)[(κxH2)3/4(κx0H2)3/4]+0.5κy11/3κH4F21(116;2;3;κH2κy2)}1/2.

Based on the integral expression [27]

02πexp[inφ1+ηcos(φ1φ2)]dφ1=2πexp(inφ2)In(η),
where In(η) is the Bessel function of the second kind with norder. By combining Eqs. (12), (16), and (19), we obtain the radial distribution functions of the signal mode probability density |βm0(r,z)|2 as m=m0 and crosstalk probability density |βm(r,z)|2 as mm0

|βm(r,z)|2=k2z2ω02(r0ω0a2)2exp(2a33)|Jm0(krr0z)|2exp(2r2ρ0xy2)Imm0(2r2ρ0xy2).

The relative power of the spiral harmonics marked with m in the paraxial regime of light propagation is determined by [28]

pm=0|βm(r,z)|2rdrm=0|βm(r,z)|2rdr.

The received power pm0 is defined as relative power of the original OAM mode m0 in the receiver plane (i.e. m=m0) and the crosstalk power pΔm is the relative power in the receiver plane that is found to be in OAM mode m=m+0Δm [6, 29].

4. Numerical results and discussion

In this section, we will carry out a study of the spiral spectrum of Airy beams in moderate-to-strong maritime atmospheric turbulence by using the formulae derived in the previous section. Numerical calculations are performed to illustrate the effects of the inner-scale, outer-scale, diameter of the receiver and light source on the transmission of OAM of Airy beams in moderate-to-strong turbulence of maritime atmosphere.

Figure 1 shows the relationship of the received power pm0 of the detected OAM states (the probability for achieving the original OAM state m0) and the propagation distance z with several values of the wavelength. The parameters are set as m=m=01, Cn2=1014m11/3, z=1km, l0=1mm, L=01m, r0=1mm,ω=01mm, a=0.05, and D=0.05m. Turbulence of maritime atmosphere is known to vary from moderate to strong fluctuation regions by increasing the index of refraction structure parameter Cn2 and propagation distance z or decreasing wavelength λ. For simplicity, we let z vary when Cn2 is given. In this case, longer propagation distance directly corresponds to a stronger fluctuation of maritime atmospheric turbulence. At a short fixed propagation distance, the received power of Airy beams slightly fluctuates with short wavelength. This effect is caused by the distribution and spread of signal mode probability density in the radial direction as the wavelength increases. At the same time, however, a shorter wavelength corresponds to a larger wave number, which induces relatively strong scintillations. Airy beams with short wavelengths are susceptible to strong fluctuations, resulting in one benefit of a larger wavelength for OAM propagation under maritime atmospheric turbulence.

 figure: Fig. 1

Fig. 1 Received power of the detected OAM states which equals the original azimuthal index m0 of Airy beams through a maritime environment as a function of the propagation distance with different wavelengths.

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The received power of the detected OAM states pm0 and crosstalk power pΔm of Airy beams through maritime environment are depicted in Figs. 2(a) and 2(b). For practical optical communication with the OAM mode, larger OAM quantum numbers correspond to the larger radius of spiral spectrum. This indicates that the received signal carried the maximum OAM modes is limited by the radius of the confined aperture in the optical system. For a given receiving aperture radius, the received power with a smaller OAM number is higher than that with a larger OAM number, as shown in Fig. 2(a). When OAM numbers further enlarge [Fig. 2(a)], the influence of OAM number on the received power can be ignored for a fixed optical system. The crosstalk power pΔm for different Δm=mm0 is represented by the histograms in Fig. 2(b), taking parameters Cn2=1013m2/3,r0=25mm. The bars labeled with Δm=0 describe the received powers that maintain the launched mode after propagation in maritime atmospheric turbulence. The other bars Δm0 indicate that the energy spreads to other OAM states caused by turbulence, and the corresponding received power becomes noise (crosstalk power). For short propagation distances, the crosstalk power is negligible. As propagation distance increases (σR21withz=700m in middle fluctuation regions), the crosstalk power only found in the neighboring channel (Δm=±1) becomes evident. As the propagation distance further increases (σR2>>1 in strong fluctuation regions), the crosstalk power is conspicuous in all of the channels, and the received power is difficult to differentiate from noises.

 figure: Fig. 2

Fig. 2 Received power of the detected OAM states of Airy beams through maritime environment as a function of propagation distance with different azimuthal indices: (a) m=m0; (b) m0=1,Δm=4to4.

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Figure 3(a) illustrates the received power of the detected OAM states pm0 for various radii of the main ring of Airy beams. As r0 increases, the maximum intensity of Airy beams elevates for propagation [20]. As a consequence, in Fig. 3(a), the received power boosts as r0 increases for a fixed propagation distance. For r>r0 [r=0.05m in Fig. 3(a)], the Airy beams decay slowly with oscillations of Airy tails. As r0 becomes sufficiently large, the third term Jm0(krr0/z) on the right hand of Eq. (12) becomes significant. In Fig. 3(a), the oscillations become increasingly notable as r0 rises. As z increases, the amplitude of Airy beams slowly decay; concurrently, the power tends to concentrate in a small area because of the decrease of the radius of Airy beams, and leads to an increase in amplitude of the Airy beam [20]. The balance of these two effects results in the abrupt autofocusing of Airy beams, which causes the received power to significantly rise for a large r0 at about z=1km. For even larger values of z, the received powers are suppressed by the increasing fluctuation of the maritime atmosphere. Thus, the difference induced by r0 in the received power diminishes gradually over long distances. To illustrate the effect of aperture diameter D on the received power through marine environment, we plot Fig. 3(b). As aperture diameter D increases [the numerator of Eq. (23)], the total of the signal carried initially launched OAM states actually rises. Meanwhile, the denominator of Eq. (23) is constructed with primitive transmitting OAM states and crosstalk among all OAM channels. As D increases, the ratio of the detected OAM signal to noise decreases because of the decays with oscillations of Airy beams along the radial direction. Thus, the denominator of Eq. (23) increases more quickly than the numerator. As a consequence, a larger aperture diameter D leads to a lower received power [Fig. 3(b)]. These results imply that to optimize the received signal in such OAM communication, adopting an auto-tracking system is highly necessary.

 figure: Fig. 3

Fig. 3 Received power of the detected OAM states of Airy beams through maritime environment as a function of propagation distance with different variables: (a) the radius of the main ring r0; (b) diameter of the circular aperture D.

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Figures 4(a) and 4(b) reveal the effects of inner- and outer-scale on the received power. The inner- and outer-scale define the lower and upper limits, respectively, of scale sizes of eddies in the inertial range. Smaller inner scales induce the more severe degradation of the received power, as indicated in Fig. 4(a). This result is achieved because of the existence of a greater number of turbulent cells in turbulence for smaller inner scales. By comparing Figs. 4(a) and 4(b), we note that the influence of the outer-scale of turbulence on the received power is less than that of the inner-scale. This finding can be explained directly by κx0H2 and κx02 in Eq. (10). For clarity, the influences of the inner- and outer-scale on the spatial coherence radius are explored in Figs. 4(c) and 4(d). At Cn2=1012m2/3, the spatial coherence radius increases until it reaches a maximum value as the inner-scale increases; then ρ0xy decreases, saturating at a level as the inner-scale further increases. Instead, ρ0xy almost increases linearly as the inner-scale increases at Cn2=1014m2/3. In Figs. 4(c) and 4(d), we observe that the spatial coherence radius is not sensitive to the change in outer-scale. For l0/L0<0.01, the effect of the outer-scale becomes insignificant. The ratio of l0 to L0 is only larger than 0.01; the outer-scale gradually plays a role, then the larger outer-scale corresponds to the larger spatial coherence radius. These findings explain why the higher outer-scale, in Fig. 4(b), generates larger received power.

 figure: Fig. 4

Fig. 4 Received power of the detected OAM states of Airy beams through maritime environment with different values of l0 and L0: (a) L0=1m and (b) l0=1cm. Spatial coherence radius with different values of l0 and L0: (c) Cn2=1014m2/3 and (d) Cn2=1012m2/3.

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5. Conclusion

In this study, theoretical models for the spatial coherence radius were derived by utilizing the modified Rytov method when the spiral spectrum of the Airy beams through moderate-to-strong maritime atmospheric turbulence. On the basis of the developed models, we analyzed the received power and crosstalk power of the Airy beams that carried OAM through moderate-to-strong maritime atmospheric turbulence. Our results showed that the increment of the inner-scale significantly alleviates the spread of the spiral spectrum and increases the received power of OAM. By contrast, the outer-scale elicits almost no effect on the received power at l0/L0<0.01. The outer-scale began to work on the spatial coherence radius and the received power at l0/L0>0.01. The performance of the light source plays a key role in the received power of the OAM under moderate-to-strong maritime atmospheric turbulence. The spiral spectrum of the Airy beams is less affected by turbulence under longer wavelengths, smaller OAM numbers, larger radii of the main ring and smaller diameters of the circular aperture. Autofocusing of Airy beams is beneficial for the propagation of the spiral spectrum in a certain propagation distance. Our study is useful for the selection of light source and the design of an optical communication system with OAM encoding through moderate-to-strong maritime atmospheric turbulence.

Acknowledgment

This work is supported by the Natural Science Foundation of Jiangsu Province of China (Grant No. BK20140128), the Fundamental Research Funds for the Central Universities (Grant No. JUSRP51517).

References and links

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References

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  1. C. A. Friehe, J. C. La Rue, F. H. Champagne, C. H. Gibson, and G. F. Dreyer, “Effects of temperature and humidity fluctuations on the optical refractive index in the marine boundary layer,” J. Opt. Soc. Am. 65(12), 1502–1511 (1975).
    [Crossref]
  2. R. J. Hill, “Spectra of fluctuations in refractivity, temperature, humidity, and the temperature-humidity cospectrum in the inertial and dissipation ranges,” Radio Sci. 13(6), 953–961 (1978).
    [Crossref]
  3. K. J. Grayshan, F. S. Vetelino, and C. Y. Young, “A marine atmospheric spectrum for laser propagation,” Waves Random Complex Media 18(1), 173–184 (2008).
    [Crossref]
  4. F. S. Vetelino, K. Grayshan, and C. Y. Young, “Inferring path average Cn2 values in the marine environment,” J. Opt. Soc. Am. A 24(10), 3198–3206 (2007).
    [Crossref] [PubMed]
  5. M. Cheng, L. Guo, and Y. Zhang, “Scintillation and aperture averaging for Gaussian beams through non-Kolmogorov maritime atmospheric turbulence channels,” Opt. Express 23(25), 32606–32621 (2015).
    [Crossref] [PubMed]
  6. C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94(15), 153901 (2005).
    [Crossref] [PubMed]
  7. J. A. Anguita, M. A. Neifeld, and B. V. Vasic, “Turbulence-induced channel crosstalk in an orbital angular momentum-multiplexed free-space optical link,” Appl. Opt. 47(13), 2414–2429 (2008).
    [Crossref] [PubMed]
  8. J. Ou, Y. S. Jiang, J. H. Zhang, H. Tang, Y. T. He, S. H. Wang, and J. L. Liao, “Spreading of spiral spectrum of Bessel–Gaussian beam in non-Kolmogorov turbulence,” Opt. Commun. 318, 95–99 (2014).
    [Crossref]
  9. Y. Zhu, L. Zhang, Z. Hu, and Y. Zhang, “Effects of non-Kolmogorov turbulence on the spiral spectrum of Hypergeometric-Gaussian laser beams,” Opt. Express 23(7), 9137–9146 (2015).
    [Crossref] [PubMed]
  10. Y. S. Jiang, S. H. Wang, J. H. Zhang, J. Ou, and H. Tang, “Spiral spectrum of Laguerre–Gaussian beam propagation in non-Kolmogorov turbulence,” Opt. Commun. 303, 38–41 (2013).
    [Crossref]
  11. L. Allen, M. W. Beijersbergen, R. J. 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]
  12. G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12(22), 5448–5456 (2004).
    [Crossref] [PubMed]
  13. Z. Bouchal and R. Celechovský, “Mixed vortex states of light as information carriers,” New J. Phys. 6, 131 (2004).
    [Crossref]
  14. B. Rodenburg, M. P. Lavery, M. Malik, M. N. O’Sullivan, M. Mirhosseini, D. J. Robertson, M. Padgett, and R. W. Boyd, “Influence of atmospheric turbulence on states of light carrying orbital angular momentum,” Opt. Lett. 37(17), 3735–3737 (2012).
    [Crossref] [PubMed]
  15. Y. Zhang, M. Cheng, Y. Zhu, J. Gao, W. Dan, Z. Hu, and F. Zhao, “Influence of atmospheric turbulence on the transmission of orbital angular momentum for Whittaker-Gaussian laser beams,” Opt. Express 22(18), 22101–22110 (2014).
    [Crossref] [PubMed]
  16. Y. Zhu, X. Liu, J. Gao, Y. Zhang, and F. Zhao, “Probability density of the orbital angular momentum mode of Hankel-Bessel beams in an atmospheric turbulence,” Opt. Express 22(7), 7765–7772 (2014).
    [Crossref] [PubMed]
  17. M. V. Berry and N. L. Baazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979).
    [Crossref]
  18. J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16(17), 12880–12891 (2008).
    [Crossref] [PubMed]
  19. X. Chu, G. Zhou, and R. Chen, “Analytical study of the self-healing property of Airy beams,” Phys. Rev. A 85(1), 013815 (2012).
    [Crossref]
  20. N. K. Efremidis and D. N. Christodoulides, “Abruptly autofocusing waves,” Opt. Lett. 35(23), 4045–4047 (2010).
    [Crossref] [PubMed]
  21. P. Zhang, J. Prakash, Z. Zhang, M. S. Mills, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, “Trapping and guiding microparticles with morphing autofocusing Airy beams,” Opt. Lett. 36(15), 2883–2885 (2011).
    [Crossref] [PubMed]
  22. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press, 2001).
  23. J. E. Morris, M. Mazilu, J. Baumgartl, T. Cizmár, and K. Dholakia, “Propagation characteristics of Airy beams: dependence upon spatial coherence and wavelength,” Opt. Express 17(15), 13236–13245 (2009).
    [Crossref] [PubMed]
  24. G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88(1), 013601 (2001).
    [Crossref] [PubMed]
  25. 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]
  26. H. Xu, Z. Cui, and J. Qu, “Propagation of elegant Laguerre-Gaussian beam in non-Kolmogorov turbulence,” Opt. Express 19(22), 21163–21173 (2011).
    [Crossref] [PubMed]
  27. A. Jeffrey and D. Zwillinger, Table of Integrals, Series, and Products (Academic, 2007).
  28. Y. Liu, C. Gao, M. Gao, and F. Li, “Coherent-mode representation and orbital angular momentum spectrum of partially coherent beam,” Opt. Commun. 281(8), 1968–1975 (2008).
    [Crossref]
  29. G. A. Tyler and R. W. Boyd, “Influence of atmospheric turbulence on the propagation of quantum states of light carrying orbital angular momentum,” Opt. Lett. 34(2), 142–144 (2009).
    [Crossref] [PubMed]

2015 (2)

2014 (3)

2013 (1)

Y. S. Jiang, S. H. Wang, J. H. Zhang, J. Ou, and H. Tang, “Spiral spectrum of Laguerre–Gaussian beam propagation in non-Kolmogorov turbulence,” Opt. Commun. 303, 38–41 (2013).
[Crossref]

2012 (2)

2011 (2)

2010 (2)

2009 (2)

2008 (4)

Y. Liu, C. Gao, M. Gao, and F. Li, “Coherent-mode representation and orbital angular momentum spectrum of partially coherent beam,” Opt. Commun. 281(8), 1968–1975 (2008).
[Crossref]

J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16(17), 12880–12891 (2008).
[Crossref] [PubMed]

J. A. Anguita, M. A. Neifeld, and B. V. Vasic, “Turbulence-induced channel crosstalk in an orbital angular momentum-multiplexed free-space optical link,” Appl. Opt. 47(13), 2414–2429 (2008).
[Crossref] [PubMed]

K. J. Grayshan, F. S. Vetelino, and C. Y. Young, “A marine atmospheric spectrum for laser propagation,” Waves Random Complex Media 18(1), 173–184 (2008).
[Crossref]

2007 (1)

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]

2004 (2)

2001 (1)

G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88(1), 013601 (2001).
[Crossref] [PubMed]

1992 (1)

L. Allen, M. W. Beijersbergen, R. J. 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]

1979 (1)

M. V. Berry and N. L. Baazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979).
[Crossref]

1978 (1)

R. J. Hill, “Spectra of fluctuations in refractivity, temperature, humidity, and the temperature-humidity cospectrum in the inertial and dissipation ranges,” Radio Sci. 13(6), 953–961 (1978).
[Crossref]

1975 (1)

Allen, L.

L. Allen, M. W. Beijersbergen, R. J. 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]

Anguita, J. A.

Baazs, N. L.

M. V. Berry and N. L. Baazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979).
[Crossref]

Barnett, S.

Baumgartl, J.

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. 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]

Berry, M. V.

M. V. Berry and N. L. Baazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979).
[Crossref]

Bouchal, Z.

Z. Bouchal and R. Celechovský, “Mixed vortex states of light as information carriers,” New J. Phys. 6, 131 (2004).
[Crossref]

Boyd, R. W.

Broky, J.

Celechovský, R.

Z. Bouchal and R. Celechovský, “Mixed vortex states of light as information carriers,” New J. Phys. 6, 131 (2004).
[Crossref]

Champagne, F. H.

Chen, R.

X. Chu, G. Zhou, and R. Chen, “Analytical study of the self-healing property of Airy beams,” Phys. Rev. A 85(1), 013815 (2012).
[Crossref]

Chen, Z.

Cheng, M.

Christodoulides, D. N.

Chu, X.

X. Chu, G. Zhou, and R. Chen, “Analytical study of the self-healing property of Airy beams,” Phys. Rev. A 85(1), 013815 (2012).
[Crossref]

Cizmár, T.

Courtial, J.

Cui, Z.

Dan, W.

Dholakia, K.

Dogariu, A.

Dreyer, G. F.

Efremidis, N. K.

Franke-Arnold, S.

Friehe, C. A.

Gao, C.

Y. Liu, C. Gao, M. Gao, and F. Li, “Coherent-mode representation and orbital angular momentum spectrum of partially coherent beam,” Opt. Commun. 281(8), 1968–1975 (2008).
[Crossref]

Gao, J.

Gao, M.

Y. Liu, C. Gao, M. Gao, and F. Li, “Coherent-mode representation and orbital angular momentum spectrum of partially coherent beam,” Opt. Commun. 281(8), 1968–1975 (2008).
[Crossref]

Gibson, C. H.

Gibson, G.

Grayshan, K.

Grayshan, K. J.

K. J. Grayshan, F. S. Vetelino, and C. Y. Young, “A marine atmospheric spectrum for laser propagation,” Waves Random Complex Media 18(1), 173–184 (2008).
[Crossref]

Guo, H.

Guo, L.

He, Y. T.

J. Ou, Y. S. Jiang, J. H. Zhang, H. Tang, Y. T. He, S. H. Wang, and J. L. Liao, “Spreading of spiral spectrum of Bessel–Gaussian beam in non-Kolmogorov turbulence,” Opt. Commun. 318, 95–99 (2014).
[Crossref]

Hill, R. J.

R. J. Hill, “Spectra of fluctuations in refractivity, temperature, humidity, and the temperature-humidity cospectrum in the inertial and dissipation ranges,” Radio Sci. 13(6), 953–961 (1978).
[Crossref]

Hu, Z.

Jiang, Y. S.

J. Ou, Y. S. Jiang, J. H. Zhang, H. Tang, Y. T. He, S. H. Wang, and J. L. Liao, “Spreading of spiral spectrum of Bessel–Gaussian beam in non-Kolmogorov turbulence,” Opt. Commun. 318, 95–99 (2014).
[Crossref]

Y. S. Jiang, S. H. Wang, J. H. Zhang, J. Ou, and H. Tang, “Spiral spectrum of Laguerre–Gaussian beam propagation in non-Kolmogorov turbulence,” Opt. Commun. 303, 38–41 (2013).
[Crossref]

La Rue, J. C.

Lavery, M. P.

Li, F.

Y. Liu, C. Gao, M. Gao, and F. Li, “Coherent-mode representation and orbital angular momentum spectrum of partially coherent beam,” Opt. Commun. 281(8), 1968–1975 (2008).
[Crossref]

Liao, J. L.

J. Ou, Y. S. Jiang, J. H. Zhang, H. Tang, Y. T. He, S. H. Wang, and J. L. Liao, “Spreading of spiral spectrum of Bessel–Gaussian beam in non-Kolmogorov turbulence,” Opt. Commun. 318, 95–99 (2014).
[Crossref]

Liu, X.

Liu, Y.

Y. Liu, C. Gao, M. Gao, and F. Li, “Coherent-mode representation and orbital angular momentum spectrum of partially coherent beam,” Opt. Commun. 281(8), 1968–1975 (2008).
[Crossref]

Luo, B.

Malik, M.

Mazilu, M.

Mills, M. S.

Mirhosseini, M.

Molina-Terriza, G.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88(1), 013601 (2001).
[Crossref] [PubMed]

Morris, J. E.

Neifeld, M. A.

O’Sullivan, M. N.

Ou, J.

J. Ou, Y. S. Jiang, J. H. Zhang, H. Tang, Y. T. He, S. H. Wang, and J. L. Liao, “Spreading of spiral spectrum of Bessel–Gaussian beam in non-Kolmogorov turbulence,” Opt. Commun. 318, 95–99 (2014).
[Crossref]

Y. S. Jiang, S. H. Wang, J. H. Zhang, J. Ou, and H. Tang, “Spiral spectrum of Laguerre–Gaussian beam propagation in non-Kolmogorov turbulence,” Opt. Commun. 303, 38–41 (2013).
[Crossref]

Padgett, M.

Pas’ko, V.

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]

Prakash, J.

Qu, J.

Robertson, D. J.

Rodenburg, B.

Siviloglou, G. A.

Spreeuw, R. J.

L. Allen, M. W. Beijersbergen, R. J. 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]

Tang, H.

J. Ou, Y. S. Jiang, J. H. Zhang, H. Tang, Y. T. He, S. H. Wang, and J. L. Liao, “Spreading of spiral spectrum of Bessel–Gaussian beam in non-Kolmogorov turbulence,” Opt. Commun. 318, 95–99 (2014).
[Crossref]

Y. S. Jiang, S. H. Wang, J. H. Zhang, J. Ou, and H. Tang, “Spiral spectrum of Laguerre–Gaussian beam propagation in non-Kolmogorov turbulence,” Opt. Commun. 303, 38–41 (2013).
[Crossref]

Torner, L.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88(1), 013601 (2001).
[Crossref] [PubMed]

Torres, J. P.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88(1), 013601 (2001).
[Crossref] [PubMed]

Tyler, G. A.

Vasic, B. V.

Vasnetsov, M.

Vetelino, F. S.

K. J. Grayshan, F. S. Vetelino, and C. Y. Young, “A marine atmospheric spectrum for laser propagation,” Waves Random Complex Media 18(1), 173–184 (2008).
[Crossref]

F. S. Vetelino, K. Grayshan, and C. Y. Young, “Inferring path average Cn2 values in the marine environment,” J. Opt. Soc. Am. A 24(10), 3198–3206 (2007).
[Crossref] [PubMed]

Wang, S. H.

J. Ou, Y. S. Jiang, J. H. Zhang, H. Tang, Y. T. He, S. H. Wang, and J. L. Liao, “Spreading of spiral spectrum of Bessel–Gaussian beam in non-Kolmogorov turbulence,” Opt. Commun. 318, 95–99 (2014).
[Crossref]

Y. S. Jiang, S. H. Wang, J. H. Zhang, J. Ou, and H. Tang, “Spiral spectrum of Laguerre–Gaussian beam propagation in non-Kolmogorov turbulence,” Opt. Commun. 303, 38–41 (2013).
[Crossref]

Woerdman, J. P.

L. Allen, M. W. Beijersbergen, R. J. 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]

Wu, G.

Xu, H.

Young, C. Y.

K. J. Grayshan, F. S. Vetelino, and C. Y. Young, “A marine atmospheric spectrum for laser propagation,” Waves Random Complex Media 18(1), 173–184 (2008).
[Crossref]

F. S. Vetelino, K. Grayshan, and C. Y. Young, “Inferring path average Cn2 values in the marine environment,” J. Opt. Soc. Am. A 24(10), 3198–3206 (2007).
[Crossref] [PubMed]

Yu, S.

Zhang, J. H.

J. Ou, Y. S. Jiang, J. H. Zhang, H. Tang, Y. T. He, S. H. Wang, and J. L. Liao, “Spreading of spiral spectrum of Bessel–Gaussian beam in non-Kolmogorov turbulence,” Opt. Commun. 318, 95–99 (2014).
[Crossref]

Y. S. Jiang, S. H. Wang, J. H. Zhang, J. Ou, and H. Tang, “Spiral spectrum of Laguerre–Gaussian beam propagation in non-Kolmogorov turbulence,” Opt. Commun. 303, 38–41 (2013).
[Crossref]

Zhang, L.

Zhang, P.

Zhang, Y.

Zhang, Z.

Zhao, F.

Zhou, G.

X. Chu, G. Zhou, and R. Chen, “Analytical study of the self-healing property of Airy beams,” Phys. Rev. A 85(1), 013815 (2012).
[Crossref]

Zhu, Y.

Am. J. Phys. (1)

M. V. Berry and N. L. Baazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979).
[Crossref]

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

New J. Phys. (1)

Z. Bouchal and R. Celechovský, “Mixed vortex states of light as information carriers,” New J. Phys. 6, 131 (2004).
[Crossref]

Opt. Commun. (3)

J. Ou, Y. S. Jiang, J. H. Zhang, H. Tang, Y. T. He, S. H. Wang, and J. L. Liao, “Spreading of spiral spectrum of Bessel–Gaussian beam in non-Kolmogorov turbulence,” Opt. Commun. 318, 95–99 (2014).
[Crossref]

Y. S. Jiang, S. H. Wang, J. H. Zhang, J. Ou, and H. Tang, “Spiral spectrum of Laguerre–Gaussian beam propagation in non-Kolmogorov turbulence,” Opt. Commun. 303, 38–41 (2013).
[Crossref]

Y. Liu, C. Gao, M. Gao, and F. Li, “Coherent-mode representation and orbital angular momentum spectrum of partially coherent beam,” Opt. Commun. 281(8), 1968–1975 (2008).
[Crossref]

Opt. Express (8)

H. Xu, Z. Cui, and J. Qu, “Propagation of elegant Laguerre-Gaussian beam in non-Kolmogorov turbulence,” Opt. Express 19(22), 21163–21173 (2011).
[Crossref] [PubMed]

J. E. Morris, M. Mazilu, J. Baumgartl, T. Cizmár, and K. Dholakia, “Propagation characteristics of Airy beams: dependence upon spatial coherence and wavelength,” Opt. Express 17(15), 13236–13245 (2009).
[Crossref] [PubMed]

G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12(22), 5448–5456 (2004).
[Crossref] [PubMed]

Y. Zhu, L. Zhang, Z. Hu, and Y. Zhang, “Effects of non-Kolmogorov turbulence on the spiral spectrum of Hypergeometric-Gaussian laser beams,” Opt. Express 23(7), 9137–9146 (2015).
[Crossref] [PubMed]

M. Cheng, L. Guo, and Y. Zhang, “Scintillation and aperture averaging for Gaussian beams through non-Kolmogorov maritime atmospheric turbulence channels,” Opt. Express 23(25), 32606–32621 (2015).
[Crossref] [PubMed]

Y. Zhang, M. Cheng, Y. Zhu, J. Gao, W. Dan, Z. Hu, and F. Zhao, “Influence of atmospheric turbulence on the transmission of orbital angular momentum for Whittaker-Gaussian laser beams,” Opt. Express 22(18), 22101–22110 (2014).
[Crossref] [PubMed]

Y. Zhu, X. Liu, J. Gao, Y. Zhang, and F. Zhao, “Probability density of the orbital angular momentum mode of Hankel-Bessel beams in an atmospheric turbulence,” Opt. Express 22(7), 7765–7772 (2014).
[Crossref] [PubMed]

J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16(17), 12880–12891 (2008).
[Crossref] [PubMed]

Opt. Lett. (5)

Phys. Rev. A (2)

X. Chu, G. Zhou, and R. Chen, “Analytical study of the self-healing property of Airy beams,” Phys. Rev. A 85(1), 013815 (2012).
[Crossref]

L. Allen, M. W. Beijersbergen, R. J. 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. (2)

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

G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88(1), 013601 (2001).
[Crossref] [PubMed]

Radio Sci. (1)

R. J. Hill, “Spectra of fluctuations in refractivity, temperature, humidity, and the temperature-humidity cospectrum in the inertial and dissipation ranges,” Radio Sci. 13(6), 953–961 (1978).
[Crossref]

Waves Random Complex Media (1)

K. J. Grayshan, F. S. Vetelino, and C. Y. Young, “A marine atmospheric spectrum for laser propagation,” Waves Random Complex Media 18(1), 173–184 (2008).
[Crossref]

Other (2)

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press, 2001).

A. Jeffrey and D. Zwillinger, Table of Integrals, Series, and Products (Academic, 2007).

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Figures (4)

Fig. 1
Fig. 1 Received power of the detected OAM states which equals the original azimuthal index m0 of Airy beams through a maritime environment as a function of the propagation distance with different wavelengths.
Fig. 2
Fig. 2 Received power of the detected OAM states of Airy beams through maritime environment as a function of propagation distance with different azimuthal indices: (a) m = m 0 ; (b) m 0 = 1 , Δ m = 4 to 4 .
Fig. 3
Fig. 3 Received power of the detected OAM states of Airy beams through maritime environment as a function of propagation distance with different variables: (a) the radius of the main ring r 0 ; (b) diameter of the circular aperture D.
Fig. 4
Fig. 4 Received power of the detected OAM states of Airy beams through maritime environment with different values of l 0 and L 0 : (a) L 0 = 1 m and (b) l 0 = 1 cm . Spatial coherence radius with different values of l 0 and L 0 : (c) C n 2 = 10 14 m 2 / 3 and (d) C n 2 = 10 12 m 2 / 3 .

Equations (23)

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

ϕ n ( κ ) = 0.033 C n 2 κ 11 / 3 , 0 κ < ,
ϕ n , e f f ( κ ) = ϕ n ( κ ) [ f ( κ l 0 ) g ( κ L 0 ) G x ( κ ) + G y ( κ ) ] = ϕ n ( κ ) [ G x ( κ , l 0 , L 0 ) + G y ( κ ) ] ,
f ( κ l 0 ) = exp ( κ 2 κ H 2 ) [ 1 0.061 κ κ H + 2.836 ( κ κ H ) 7 / 6 ] , κ H = 3.41 l 0 ,
g ( κ L 0 ) = 1 exp ( κ 2 κ 0 2 ) ,
G x ( κ ) = exp ( κ 2 κ x 2 ) ,
G y ( κ ) = κ 11 / 3 ( κ 2 + κ y 2 ) 11 / 6 ,
η x 2.61 Q 0 / [ 2.61 + Q 0 ( 1 + 0.65 d 2 + 0.45 σ R 2 Q H 1 / 6 ) ] ,
η y = 3 ( σ R σ P ) 12 / 5 ( 1 + 0.69 σ P 12 / 5 ) ,
σ P 2 = 3.86 σ 1 2 { ( 1 + 1 Q H 2 ) 11 / 12 [ sin ( 11 6 tan 1 Q H ) 0.051 sin ( 4 3 tan 1 Q H ) ( 1 + Q H 2 ) 1 / 4 + 3.052 sin ( 5 4 tan 1 Q H ) ( 1 + Q H 2 ) 7 / 24 ] 5.581 Q H 5 / 6 } .
ϕ n , e f f ( κ ) = 0.033 C n 2 { 1 [ κ 2 + κ y 2 ] 11 / 6 + κ 11 / 3 [ 1 0.061 κ κ H + 2.836 ( κ κ H ) 7 / 6 ] [ exp ( κ 2 κ x H 2 ) exp ( κ 2 κ x 0 H 2 ) ] } ,
A i 0 ( r , φ , z = 0 ) = Ai [ ± ( r 0 r ω 0 ) ] exp [ ± a ( r 0 r ω 0 ) ] exp ( i m 0 φ ) ,
A i 0 ( r , φ , z ) = i k z ω 0 ( r 0 ω 0 a 2 ) J m 0 ( k r r 0 z ) exp ( i k r 2 2 z + a 3 3 i m 0 φ ) ,
A i ( r , φ , z ) = A i 0 ( r , φ , z ) exp [ ψ x ( r , φ , z ) + ψ y ( r , φ , z ) ] ,
A i ( r , φ , z ) = 1 2 π m β m ( r , z ) exp ( i m φ ) ,
β m ( r , z ) = 1 2 π 0 2 π A i ( r , φ , z ) exp ( i m φ ) d φ .
| β m ( r , z ) | 2 = 1 2 π 0 2 π 0 2 π A i 0 ( r , φ , z ) A i 0 ( r , φ , z ) exp [ i m ( φ φ ) ] × exp [ ψ x ( r , φ , z ) + ψ x ( r , φ , z ) + ψ y ( r , φ , z ) + ψ y ( r , φ , z ) ] d φ d φ .
exp [ ψ x ( r , φ , z ) + ψ x ( r , φ , z ) + ψ y ( r , φ , z ) + ψ y ( r , φ , z ) ] = exp { π 2 k 2 z 3 [ 2 r 2 2 r 2 cos ( φ φ ) ] 0 κ 3 ϕ n , e f f ( κ ) d κ } = exp { [ 2 r 2 2 r 2 cos ( φ φ ) ] ( 1 ρ 0 x 2 + 1 ρ 0 y 2 ) } ,
ρ 0 x 2 = π 2 k 2 z 3 0.033 C n 2 0 κ 2 / 3 G x ( κ , l 0 , L 0 ) d κ = 0.054 k 2 C n 2 z { Γ ( 1 6 ) [ ( κ x H 2 ) 1 / 6 ( κ x 0 H 2 ) 1 / 6 ] 0.061 κ H Γ ( 2 3 ) [ ( κ x H 2 ) 2 / 3 ( κ x 0 H 2 ) 2 / 3 ] + 2.836 κ H 7 / 6 Γ ( 3 4 ) [ ( κ x H 2 ) 3 / 4 ( κ x 0 H 2 ) 3 / 4 ] } , ρ 0 y 2 = π 2 k 2 z 3 0.033 C n 2 0 κ H κ 2 / 3 G y ( κ ) d κ = 0.027 k 2 z C n 2 κ y 11 / 3 κ H 4 F 2 1 ( 11 6 ; 2 ; 3 ; κ H 2 κ y 2 ) .
exp [ ψ x ( r , φ , z ) + ψ x ( r , φ , z ) + ψ y ( r , φ , z ) + ψ y ( r , φ , z ) ] = exp [ 2 r 2 2 r 2 cos ( φ φ ) ρ 0 x y 2 ] ,
ρ 0 x y = ( 0.054 C n 2 k 2 z ) 1 / 2 × { Γ ( 1 6 ) [ ( κ x H 2 ) 1 / 6 ( κ x 0 H 2 ) 1 / 6 ] 0.061 κ H Γ ( 2 3 ) [ ( κ x H 2 ) 2 / 3 ( κ x 0 H 2 ) 2 / 3 ] + 2.836 κ H 7 / 6 Γ ( 3 4 ) [ ( κ x H 2 ) 3 / 4 ( κ x 0 H 2 ) 3 / 4 ] + 0.5 κ y 11 / 3 κ H 4 F 2 1 ( 11 6 ; 2 ; 3 ; κ H 2 κ y 2 ) } 1 / 2 .
0 2 π exp [ i n φ 1 + η cos ( φ 1 φ 2 ) ] d φ 1 = 2 π exp ( i n φ 2 ) I n ( η ) ,
| β m ( r , z ) | 2 = k 2 z 2 ω 0 2 ( r 0 ω 0 a 2 ) 2 exp ( 2 a 3 3 ) | J m 0 ( k r r 0 z ) | 2 exp ( 2 r 2 ρ 0xy 2 ) I m m 0 ( 2 r 2 ρ 0 x y 2 ) .
p m = 0 | β m ( r , z ) | 2 r d r m = 0 | β m ( r , z ) | 2 r d r .

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