Intensity of vortex modes carried by Lommel beam in weak-to-strong non-Kolmogorov turbulence

Lin Yu; Beibei Hu; Yixin Zhang

doi:10.1364/OE.25.019538

1. Introduction

Free-space optical communications based on various vortex beams have attracted a lot of attention in recent years because of their high channel capacity [1,2]. In addition to amplitude, phase, frequency and polarization, the mode of vortex beam carries orbital angular momentum (OAM), which provides a new degree of freedom for information encoding [3]. Since the vortex modes with different OAM quantum numbers are orthogonal, vortex beams are widely used in mode division multiplexing. However, the beam’s wavefront distortions caused by atmospheric turbulence always induce the modal crosstalk and increase the bit error rate [4]. The influence of turbulence effects on the propagation of vortex mode has been analyzed in some pervious reports, mainly focused on the classical Laguerre-Gaussian (LG) beam [5,6]. Further studies reveal that some nondiffracting vortex beams, such as Bessel-Gaussian beam [7], Hankel-Bessel beam [8], Airy beam [9], may be good alternatives to LG beam. The transverse intensity profiles of nondiffracting beams are structurally preserved upon propagation and able to reconstruct after encountering obstacles, which help to mitigate the adverse effects of turbulence [10].

As a new type of nondiffracting beam, Lommel beam is essentially an infinite linear superposition of Bessel modes whose wave vectors have identical axial projections [11]. In contrast with the radial symmetry of Bessel modes, the intensity profile of Lommel beam is symmetric about the Cartesian coordinate axes. The symmetry can be adjusted conveniently by only one parameter. Since proposed by Kovalev and Kotlyar, Lommel beam attracted the academic concern rapidly. Belafhal et al. analyzed its scattering properties by a rigid and isolated sphere for applications such as optical trapping [12]. Ez-zariy et al. simulated the axial intensity of a Lommel modulated Gaussian beam propagating in turbulence [13]. Zhao et al. made the first experimental realization of Lommel beam by use of binary amplitude masks [14]. However, the OAM propagation properties of Lommel beam in turbulence haven’t been investigated yet. Considering its specialty of continuous changeable OAM [11], Lommel beam is more likely to realize the essential mode division multiplexing, and is worthy of research in further details.

In this paper, we discuss the turbulence effects on the propagation of vortex modes for Lommel beam. The effective power spectrum model of non-Kolmogorov atmospheric turbulence in weak-to-strong region is established by use of the spatial filter. The analytic expressions are derived to analyze the received intensity of vortex modes including signal and crosstalk. Finally, numerical simulations are used to demonstrate the relationships between the intensity of vortex modes and the parameters of beam or turbulence, so as to obtain the optimal parameters of free-space OAM communication using Lommel beam.

2. Theoretical model

In the cylindrical coordinate system, the electric field of Lommel beam is expressed as [11]

E_{0} (r, φ, z) = c^{- n_{0}} \exp (i z \sqrt{k^{2} - k_{ρ}^{2}}) u_{n_{0}} [c k_{ρ} r \exp (i φ), k_{ρ} r],

where

z

is the propagation distance,

r

and

φ

are the radial and angular coordinate respectively in the

z

plane,

k_{ρ}

is the transverse component of the beam’s wavenumber

k

(

k = 2 π / λ

), the variables

c

,

λ

and

n_{0}

are the beam’s asymmetry parameter, wavelength and OAM quantum number respectively. To ensure the convergence of Lommel function

u_{n_{0}} [c k_{ρ} r \exp (i φ), k_{ρ} r]

, the modulus of

c

should be less than unity. If expanding

u_{n_{0}}

in the paraxial case (

k_{ρ} ≪ k

), Eq. (1) can be rewritten as

E_{0} (r, φ, z) = \exp (i k z) \sum_{p = 0}^{\infty} {(- c^{2})}^{p} \exp [i (n_{0} + 2 p) φ] J_{n_{0} + 2 p} (k_{ρ} r) .

Figure 1 gives the transverse intensity distribution of Lommel beam at the source plane ( $z = 0$ ). The beam parameters are set as $n_{0} = 1$ , $k_{ρ} = 0.001 k$ , $λ = 1550 nm$ and the coordinates are limited in the range $- 0.01 m \leq x, y \leq 0.01 m$ . Obviously, the asymmetry parameter $c$ determines the shape of intensity profile in Fig. 1, and can be used for beam control. With the increment of the modulus of $c$ , the circular symmetry gradually degenerates into two crescents with axial symmetry. The direction of symmetric axis is decided by the argument of $c$ . In particular, as for pure real and imaginary number of $c$ , the intensity profile of Lommel beam is symmetric about $x$ axis and $y$ axis respectively.

Fig. 1 The transverse intensity pattern of Lommel beam with different asymmetry parameters at the source plane.

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When Lommel beam propagates in atmospheric turbulence, the refractive index fluctuation of atmosphere contributes an additional complex phase perturbation factor $ψ (r, φ, z)$ to $E_{0}$ , which results in the deviation of vortex modes from the original OAM eigenstates. In order to obtain the weight of new vortex mode component, the disturbed electric field $E$ is decomposed into a series of spiral harmonics $\exp (i n φ)$ with corresponding coefficient $β_{n}$ as follows [15]:

E (r, φ, z) = E_{0} (r, φ, z) \exp [ψ (r, φ, z)] = \frac{1}{\sqrt{2 π}} \sum_{n = - \infty}^{\infty} β_{n} \exp (i n φ),

where

β_{n}

is given by the integral

β_{n} = \frac{1}{\sqrt{2 π}} \int_{0}^{2 π} E (r, φ, z) \exp (- i n φ) d φ .

The ensemble average of

{| β_{n} |}^{2}

represents the probability density of vortex mode in atmospheric turbulence, which has the form of

〈 {| β_{n} |}^{2} 〉 = \frac{1}{2 π} \int_{0}^{2 π} \int_{0}^{2 π} E_{0} (r, φ, z) E_{0}^{*} (r, φ^{'}, z) 〈 \exp [ψ (r, φ, z) + ψ^{*} (r, φ^{'}, z)] 〉 \exp [- i n (φ - φ^{'})] d φ^{'} d φ .

According to the definition of complex phase structure function [16], we can get

\begin{array}{l} 〈 \exp [ψ (r, φ, z) + ψ^{*} (r, φ', z)] 〉 = \exp [- \frac{1}{3} π^{2} k^{2} z [2 r^{2} - 2 r^{2} \cos (φ - φ')] \int_{0}^{\infty} κ^{3} Φ_{n e f f} (κ) d κ], \\ = \exp [- \frac{2 r^{2} - 2 r^{2} \cos (φ - φ^{'})}{ρ_{0}^{2}}] \end{array}

where

ρ_{0}

is the spatial coherence radius,

κ

and

Φ_{n e f f}

are the spatial frequency and the effective power spectrum of refractive index fluctuations respectively. Substituting Eq. (2) and Eq. (6) into Eq. (5), and simplifying the expression by some integral calculations, the final result of mode probability density can be written as

\begin{array}{l} 〈 {| β_{n} |}^{2} 〉 = \frac{1}{2 π} \sum_{p^{'} = 0}^{\infty} \sum_{p = 0}^{\infty} {(- 1)}^{p + p^{'}} \int_{0}^{2 π} \int_{0}^{2 π} c^{2 p + 2 p^{'}} \exp [i 2 (p - p^{'}) φ] J_{n_{0} + 2 p} (k_{ρ} r) J_{n_{0} + 2 p^{'}} (k_{ρ} r) \\ \times \exp [- \frac{2 r^{2}}{ρ_{0}^{2}}] \exp [\frac{2 r^{2} \cos (φ - φ^{'})}{ρ_{0}^{2}} - i (n - n_{0} - 2 p^{'}) (φ - φ^{'})] d φ^{'} d φ, \\ = 2 π \sum_{p = 0}^{\infty} c^{4 p} {[J_{n_{0} + 2 p} (k_{ρ} r)]}^{2} \exp [- \frac{2 r^{2}}{ρ_{0}^{2}}] I_{n - n_{0} - 2 p} (\frac{2 r^{2}}{ρ_{0}^{2}}) \end{array}

where

J_{n_{0} + 2 p}

and

I_{n - n_{0} - 2 p}

denote Bessel function of the first kind and the modified Bessel function of the first kind respectively. The difference between

n

and

n_{0}

determines whether the received vortex mode is a signal mode or a crosstalk mode. If we define

Δ n = n - n_{0}

, Eq. (7) is transformed into

〈 {| β_{Δ n} |}^{2} 〉 = 2 π \sum_{p = 0}^{\infty} c^{4 p} {[J_{n_{0} + 2 p} (k_{ρ} r)]}^{2} \exp [- \frac{2 r^{2}}{ρ_{0}^{2}}] I_{Δ n - 2 p} (\frac{2 r^{2}}{ρ_{0}^{2}}) .

The probability density of signal mode (

Δ n = 0

) and crosstalk mode (

Δ n \neq 0

) can then be calculated using Eq. (8). As for a finite-aperture receiver with diameter

D

, the relative received intensity

P_{Δ n}

of vortex mode is then given by

P_{Δ n} = \frac{\int_{0}^{D / 2} 〈 {| β_{Δ n} |}^{2} 〉 r d r}{\sum_{Δ n = - \infty}^{\infty} \int_{0}^{D / 2} 〈 {| β_{Δ n} |}^{2} 〉 r d r} .

According to Eq. (8) and Eq. (9), it is noticeable that

ρ_{0}

is a major factor determining the propagation characteristics of vortex mode in atmospheric turbulence. To obtain the value of

ρ_{0}

, the key problem lies in constructing the turbulence model to describe

Φ_{n e f f}

in Eq. (6).

Here the extended Rytov approximation is used to construct the non-Kolmogorov turbulence model in weak-to-strong turbulence region, with consideration of both its inner-scale and outer-scale effects. By introducing the spatial filter functions, $Φ_{n e f f}$ can be written as

\begin{array}{l} Φ_{n e f f} (κ) = A (α) C_{n}^{2} κ^{- α} [f (κ, α, l_{0}) g (κ, L_{0}) G_{X} (κ, α) + G_{Y} (κ, α)] \\ A (α) = \frac{Γ (α - 1)}{4 π^{2}} \cos (\frac{π}{2} α), \end{array}

where

α

is the non-Kolmogorov spectrum parameter with value in the region from 3 to 4,

C_{n}^{2}

is the refractive-index structure constant with unit

m^{3 - α}

,

Γ (α - 1)

denotes the Gamma function. The terms

f (κ, α, l_{0})

and

g (κ, L_{0})

describe the inner-scale and outer-scale effect of turbulence respectively with corresponding scale factor

l_{0}

and

L_{0}

, while

G_{X} (κ, α)

and

G_{Y} (κ, α)

represent the large-scale and small-scale spatial filter function respectively. Their expressions are given by [17]

\begin{array}{l} f (κ, α, l_{0}) = \exp (- \frac{κ^{2}}{κ_{l}^{2}}), g (κ, L_{0}) = 1 - \exp (- \frac{κ^{2}}{κ_{L}^{2}}) . \\ G_{x} (κ, α) = e x p [- \frac{κ^{2}}{κ_{x}^{2}}], G_{y} (κ, α) = \frac{κ^{α}}{{[κ^{2} + κ_{y}^{2}]}^{α / 2}} \end{array}

In Eq. (11),

κ_{l} = c (α) / l_{0}

and

κ_{L} = 8 π / L_{0}

are the spatial frequency corresponding to the inner-scale and outer-scale of turbulence eddies respectively,

κ_{x} = \sqrt{k η_{x} / z}

and

κ_{y} = \sqrt{k η_{y} / z}

are the spatial cutoff frequency for large-scale and small-scale filter function respectively. The parameters

η_{x}

,

η_{y}

and

c (α)

can be further expressed as [18]

\begin{array}{l} c (α) = {[\frac{2 π}{3} A (α) Γ (\frac{5 - α}{2})]}^{1 / (α - 5)}, β (α) = - 4 Γ (1 - \frac{α}{2}) \sin \frac{π α}{4} \frac{Γ^{2} (α / 2)}{Γ (α)}, \\ η_{x} = \frac{1}{1 + f_{x} (α) σ_{R}^{4 / (α - 2)}} {[\frac{7.35 β (α)}{Γ (3 - α / 2)}]}^{2 / (6 - α)}, f_{x} (α) = {[1.02 r (α) I (α)]}^{2 / (α - 6)}, \\ η_{y} = {[0.06375 (α - 2) β (α)]}^{2 / (2 - α)} [1 + f_{y} (α) σ_{R}^{4 / (α - 2)}], f_{y} (α) = {(\frac{\ln 2}{0.51})}^{2 / (2 - α)}, \\ σ_{R}^{2} = β (α) A (α) C_{n}^{2} π^{2} k^{3 - α / 2} z^{α / 2}, I (α) = {(α - 1)}^{\frac{6 - α}{α - 2}} \frac{Γ^{2} (α - 3)}{Γ (2 α - 6)}, \\ r (α) = \frac{1}{α - 2} 2^{\frac{(3 - α) (α - 10)}{α - 2}} {[- \frac{Γ (1 - α / 2)}{Γ (α / 2)}]}^{\frac{α - 6}{α - 2}} Γ (\frac{6 - α}{α - 2}) {[β (α)]}^{\frac{8 - 2 α}{α - 2}} . \end{array}

Substituting Eq. (11) and Eq. (12) into Eq. (10), the final expression of

Φ_{n e f f}

is written as

\begin{array}{l} Φ_{n e f f} (κ) = A (α) C_{n}^{2} κ^{- α} {\exp (- \frac{κ^{2}}{κ_{l x}^{2}}) - \exp (- \frac{κ^{2}}{κ_{L l x}^{2}}) + \frac{κ^{α}}{{[κ^{2} + κ_{y}^{2}]}^{α / 2}}}, \\ \frac{1}{κ_{l x}^{2}} = \frac{1}{κ_{l}^{2}} + \frac{1}{κ_{x}^{2}}, \frac{1}{κ_{L l x}^{2}} = \frac{1}{κ_{L}^{2}} + \frac{1}{κ_{l}^{2}} + \frac{1}{κ_{x}^{2}} . \end{array}

Thus according to Eq. (6) and Eq. (13),

ρ_{0}

has the form of

\begin{array}{l} ρ_{0} = {[\frac{1}{3} π^{2} k^{2} z \int_{0}^{\infty} κ^{3} Φ_{n e f f} (κ) d κ]}^{- \frac{1}{2}} \\ = {\frac{Γ (α - 1) C_{n}^{2} k^{2} z}{48} \cos (\frac{π α}{2}) {[2 Γ (2 - \frac{α}{2}) (κ_{l x}^{4 - α} - κ_{L l x}^{4 - α}) + κ_{l}^{4} κ_{y}^{- α} {}_{2}F_{1} (\frac{α}{2}, 2; 3, - \frac{κ_{l}^{2}}{κ_{y}^{2}})]}}^{- \frac{1}{2}} . \end{array}

where

{}_{2}F_{1}

denotes the confluent hypergeometric function. Then the received vortex mode’s intensity

P_{Δ n}

of Lommel beam can be calculated by combining Eq. (8), Eq. (9) and Eq. (14), so as to analyze the turbulence effects on the propagation of vortex mode quantitatively.

3. Simulation and analysis

In this section, numerical simulations are used to demonstrate the vortex mode intensity of signal and crosstalk under different beam parameters and turbulence conditions. Unless otherwise mentioned, the parameters used in simulations are set as $n_{0} = 1$ , $c = 0.1$ , $α = 3.67$ , $λ = 1550 nm$ , $l_{0} = 1 mm$ , $L_{0} = 1 m$ , $D = 0.05 m$ and $z = 1 km .$ The intensity distributions of the received vortex modes carried by Lommel beam in weak-to-strong turbulence are shown in Fig. 2(a) to Fig. 2(c). Obviously the turbulence causes the energy migration from the initial OAM eigenstate to others. When the turbulence is weak, the crosstalk occurs mainly between adjacent modes (i.e. the difference of OAM quantum number $Δ n = \pm 1$ ) with negligible intensity. While in strong turbulence, the crosstalk intensity increases greatly and spreads to more peripheral modes. As a comparison, the received mode intensities of Laguerre-Gaussian (LG) beam under the same turbulence conditions are shown in Fig. 2(d) to Fig. 2(f). The waist radius, radial index and azimuthal index (i.e. OAM quantum number) of LG beam are set as 0.01 m, 0 and 1 respectively. No matter in weak or strong turbulence, the received signal mode intensity of LG beam is always lower than that of Lommel beam. Moreover, the total energy of LG beam tends to distribute equally among the signal mode and each crosstalk mode in strong turbulence, which is more likely to cause the severe performance degeneration or even failure of OAM multiplexing communication. So from the view of anti-turbulence interference, Lommel beam is superior to the traditional LG beam.

Fig. 2 The received mode intensity distribution of Lommel beam and Laguerre-Gaussian beam in (a, d) weak, (b, e) moderate and (c, f) strong turbulence. (a)-(c): Lommel beam; (d)-(f): Laguerre-Gaussian beam.

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Figure 3 displays the received vortex mode intensity of Lommel beam under different propagation distances and turbulent strengths. With the increment of propagation distance $z$ , the signal mode ( $Δ n = 0$ ) intensity $P_{s}$ keeps decreasing due to the accumulation of turbulence effect. Since the variation trend of the total crosstalk intensity is just contrary to $P_{s}$ without the need for further study, we discuss the chief crosstalk mode ( $Δ n = 1$ ) intensity $P_{c}$ instead in the following text. In weak turbulence ( $C_{n}^{2} {=10}^{- 15} m^{3 - α}$ ) and moderate turbulence ( $C_{n}^{2} {=10}^{- 14} m^{3 - α}$ ), $P_{c}$ is monotone increasing. As for strong turbulence ( $C_{n}^{2} {=10}^{- 13} m^{3 - α}$ ), $P_{c}$ gradually decreases after reaching its maximum, because the enhancement of high-order crosstalk ( $| Δ n | >1$ ) reduces the weight of $P_{c}$ . Since nondiffracting property is the key to mitigating the turbulence effects and maintaining high signal intensity, a long nondiffracting region is important for long distance communication. Aruga et al. used a telescope to form a distorted concave spherical wave front, and thus obtained a nondiffracting region up to a few kilometers for Bessel beam [19]. Birch et al. proposed immersing axicon in an index-matching material to reduce the radial wave vector of Bessel beam to low values, so the nondiffracting region could reach to tens of kilometers [20]. Given that Lommel beam is essentially a linear superposition of Bessel modes and their structural similarity, using the above methods to generate long-range nondiffracting Lommel beam can also be expected.

Fig. 3 The received (a) signal intensity $P_{s}$ and (b) crosstalk intensity $P_{c}$ under different propagation distances and turbulence strengths.

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To optimize the parameters of Lommel beam propagating in atmospheric turbulence, the relationships between the received mode intensity and the modulus of asymmetry parameter $c$ under different OAM quantum numbers $n_{0}$ are shown in Fig. 4. No matter in weak or strong turbulence, $P_{s}$ keeps decreasing with the increment of $| c |$ , while $P_{c}$ first increases slightly and then decreases rapidly. This is because for a large value of $| c |$ , the high-order modal crosstalk ( $| Δ n | >1$ ) becomes obvious which in turn decreases the weight of low-order crosstalk $P_{c}$ . The variation of $n_{0}$ has little influence on $P_{s}$ and $P_{c}$ in weak turbulence. However, in conditions of strong turbulence, a small $n_{0}$ can obtain a slightly higher $P_{s}$ and lower $P_{c}$ . So the values of $| c |$ and $n_{0}$ should be selected small in Lommel beam based communication to mitigate the turbulence induced crosstalk and obtain high signal intensity.

Fig. 4 In (a, b) weak and (c, d) strong turbulence, the received signal intensity $P_{s}$ and crosstalk intensity $P_{c}$ versus the OAM quantum number and the modulus of beam asymmetry parameter.

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Figure 5 shows the influence of beam wavelength $λ$ on the signal intensity $P_{s}$ and crosstalk intensity $P_{c}$ . Five typical wavelengths are selected and the range of $C_{n}^{2}$ covers the weak-to-strong turbulence. $P_{s}$ decreases with the increment of turbulence strength and the signal mode with a longer wavelength suffers less intensity loss. In conditions of long wavelength, $P_{c}$ has a contrary variation trend to $P_{s}$ because the low-order crosstalk ( $| Δ n | =1$ ) dominates the whole crosstalk. However, as for short wavelength, the significant enhancement of other crosstalk modes ( $| Δ n | >1$ ) finally cause the decrease of $P_{c}$ in strong turbulence. So the long-wavelength Lommel beam is more suitable for OAM communication if considering its less susceptibility to turbulence.

Fig. 5 The received (a) signal intensity $P_{s}$ and (b) crosstalk intensity $P_{c}$ under different beam wavelengths and turbulence strengths.

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To obtain the propagation performance of Lommel beam under different turbulent conditions, the impact of non-Kolmogorov spectrum parameter $α$ on $P_{s}$ and $P_{c}$ are investigated. Figure 6 shows that a larger $α$ corresponds to a higher $P_{s}$ , which can be attributed to the beam scintillation effect. In general, the turbulence eddies with large wavenumber induce relatively strong scintillation. Since those eddies are fewer in turbulence when $α$ moves towards 4, less scintillation and higher $P_{s}$ are achieved [21]. With the increment of $C_{n}^{2}$ , the curve’s inflection point of $P_{c}$ occurs earlier in the turbulence with small $α$ . This is unfavorable to channel multiplexing, meaning that the crosstalk is easier to spread from adjacent modes to other peripheral modes. Given that $α$ is relevant to the atmospheric layer altitude [22], an appropriate altitude selected for communication may help to reduce the crosstalk.

Fig. 6 The received (a) signal intensity $P_{s}$ and (b) crosstalk intensity $P_{c}$ under different values of non-Kolmogorov spectrum parameter and turbulence strengths.

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Fig. 7 shows the relationships between $P_{s}$ , $P_{c}$ and the turbulence scale $l_{0}$ , $L_{0}$ . No matter in weak ( $C_{n}^{2} {=10}^{- 15} m^{3 - α}$ ) or strong ( $C_{n}^{2} {=10}^{- 13} m^{3 - α}$ ) turbulence, $P_{s}$ decreases if the inner-scale factor $l_{0}$ decreases or the outer-scale factor L₀ increases, partly owing to the increment of the effective turbulent eddies’ number within the scale range [ $l_{0}$ , $L_{0}$ ]. On the other hand, the relatively large beam wander caused by a large L₀ induces the beam pointing error, which further decreases $P_{s}$ [23]. The variation trends of $P_{c}$ with $l_{0}$ and $L_{0}$ are contrary to those of $P_{s}$ in weak turbulence. However, this rule is not satisfied in strong turbulence with small $l_{0}$ or large $L_{0}$ . The whole crosstalk energy, originally concentrated on OAM modes with $| Δ n | =1$ , starts improving injection into other modes ( $| Δ n | >1$ ). So $P_{c}$ begins to decrease when $l_{0}$ is smaller or $L_{0}$ is larger than some certain values. Overall, compared with $L_{0}$ , the effects of $l_{0}$ on $P_{s}$ and $P_{c}$ are more obvious.

Fig. 7 In (a, b) weak and (c, d) strong turbulence, the received signal intensity $P_{s}$ and crosstalk intensity $P_{c}$ versus the turbulence’s inner-scale factor and outer-scale factor.

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4. Conclusion

We develop a theoretical model to investigate the vortex mode intensity of Lommel beam propagating in atmospheric turbulence. Based on the extended Rytov approximation and spatial filter functions, the model is effective for non-Kolmogorov turbulence in weak-to-strong region. With the increment of propagation distance and turbulence strength, the received signal mode intensity $P_{s}$ decreases and the OAM crosstalk spreads from adjacent modes to peripheral modes. A smaller beam asymmetry parameter, a lower OAM quantum number and a longer wavelength are helpful to improve $P_{s}$ . The influences of turbulence parameters on $P_{s}$ mainly include two aspects. The decrease of non-Kolmogorov spectrum parameter and inner-scale factor, or the increment of outer-scale factor, can cause the reduction of $P_{s}$ . The effect of inner-scale is more obvious than outer-scale. The results provide useful reference for the optimal design of OAM communication in both weak and strong atmospheric turbulence.

Funding

The Fundamental Research Funds for the Central Universities of China (JUSRP11721, JUSRP51721B, JUSRP51716A).

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Intensity of vortex modes carried by Lommel beam in weak-to-strong non-Kolmogorov turbulence

Abstract

1. Introduction

2. Theoretical model

3. Simulation and analysis

4. Conclusion

Funding

References and links

Cited By

Figures (7)

Equations (14)

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