Effects of moderate to strong turbulence on the Hankel-Bessel-Gaussian pulse beam with orbital angular momentum in the marine-atmosphere

Yixin Zhang; Lei Shan; Ye Li; Lin Yu

doi:10.1364/OE.25.033469

1. Introduction

Atmospheric temporal broadening and atmospheric scintillation of light pulse restrict the performance of free-space optical (FSO) communication links and lidar. In recent years, optical pulse propagating through turbulent media becomes a great interest issue. Liu and Yeh [1] studied the effects of spreading and wander on pulse broadening under different scattering conditions in random media and found that the dominant factor in broadening is pulse wandering in weak scattering. However, as the scattering becomes stronger, pulse broadening and intensity fluctuation will saturate. Young et al. [2] developed an analytical expression for the temporal broadening of an ultrashort collimated Gaussian pulse in a near-filed, weakly turbulent, horizontal path and showed that the pulses exhibit considerable broadening in the upper-atmosphere. Meanwhile, at ground level, the broadening can be predicted as long as the pulse is around 1 ps. Andrews et al. [3] showed that both the temporal broadening and scintillation depend mainly on the strength and outer scale of the turbulence. The temporal broadening is identical and the temporal scintillation index exhibits similar behavior regardless of whether the pulse travels in a horizontal, vertical, or angled path. Utilizing the analysis in [3] and taking into account that a Gaussian filter is ideal, Jurado-Navas et al. [4] discussed temporal broadening of optical pulse employing a Bessel filter. For the spherical-wave Gaussian and collimated space-time Gaussian pulses, Chen et al. [5] showed that the temporal broadening of both spherical-wave Gaussian and collimated space-time Gaussian pulses depend heavily on the general spectral index of the spatial power spectrum of refractive-index fluctuations in atmospheric turbulence. The temporal broadening of the optical axis of spherical-wave Gaussian pulse can also be approximately calculated using collimated space-time Gaussian pulse with the same turbulence parameters and propagation distances. Chen et al. [6] found that the increment of mean square temporal pulse width in strong, anisotropic atmospheric turbulence is proportional to the effective anisotropic factor in most situations of interest. In oceanic turbulence, Wang et al. [7] showed that the on-axis relative pulse broadening decreases as the initial pulse half-width increases. Broadening increases as the propagation length, wavelength and initial Gaussian beam radius increase.

As the activities of mankind increase off the coast, FSO communication over the maritime atmosphere is a formidable candidate for information transfer for various purposes (such as the vessel-to-vessel or ship-to-satellite communications) [8–10]. Moreover, FSO communication using orbital angular momentum (OAM) modes can greatly increase the information capacity of the communication system [11–23]. Recently, Zhu et al. [19] and Zhang et al. [20] showed the non-diffractive Airy beams and Whittaker-Gaussian beams can reduce the transition probability of photons from the OAM signal mode to adjacent modes, especially for adjacent modes with energy levels far from the signal mode. Could there be a new role for the non-diffracting pulse beam with OAM in turbulence? It is a useful issue to study. However, to the best of our knowledge, there is almost no discussion of the effects of non-diffracting vortex beam, such as the Hankel-Bessel pulse (HBP) beam, on the pulse broadening of the HBP in marine-atmosphere.

In this paper, we report the effects of the non-diffracting HBP beam and the moderate-to-strong scintillation of marine-atmosphere on pulse broadening and average intensity of the signal and crosstalk between OAM modes in the paraxial channel.

2. Average intensity of OAM modes

Under the modified Rytov approximation [9], the field of an optical pulse at the receiving plane $z$ can be represented by [24, 25]

u_{m} (r, φ, z; t) = u_{m_{0}} (r, φ, z) f (t) \exp [ψ_{x} (r, φ, z, ω) + ψ_{y} (r, φ, z, ω)],

where

z

is the propagation distance;

r = | r |,

r = (x, y)

is the two-dimensional position vector;

φ

is the azimuthal angle;

ψ_{x} (r, φ, z, ω)

and

ψ_{y} (r, φ, z, ω)

are the complex phase perturbations due to large-scale or small-scale turbulence eddies and pulse dispersion caused by turbulence respectively;

u_{m_{0}}^{} (r, φ, z)

is the normalized HBP state in the absence of turbulence at

z

plane;

f (t) =exp (- t^{2} / τ_{0}^{2})

is the Gauss pulse factor [24];

τ_{0}

is the half-width of the pulse defined by the 1/e point and is called the pulse duration.

In the paraxial channel, the traveling scalar wave $u_{m_{0}}^{} (r, φ, z)$ has the form [26]

u_{m_{0}} (r, φ, z; t) = i^{3 m_{0} + 1} m_{0}! A_{0} \sqrt{\frac{π}{2 k z}} J_{m_{0} / 2} (k r^{2} / 4 z) \exp [i (k z - \frac{π m_{0}}{4} - \frac{π}{4}) +i m_{0} φ - \frac{t^{2}}{τ_{0}^{2}}],

where

u_{m_{0}} (r, φ, z; t) = u_{m_{0}} (r, φ, z) f (t)

;

i

is the complex symbol;

m_{0}

is the OAM quantum number of the signal which describes the helical structure of the wave front around a wave front singularity;

A_{0}

is a constant that characterizes the beam power;

J_{m_{0} / 2}

is the Bessel function of integer or half-integer orders;

k = 2 π / λ

is the wave number; and

λ

is the wavelength.

In turbulent media, the refractive index fluctuations disturb the complex amplitude of the HBP state, which is no longer guaranteed to be the original OAM state. However, using the Fourier transform theory, the complex amplitude of the HBP state $u_{m} (r, φ, z, t)$ can be written as a superposition of the plane spiral waves with new OAM quantum number $m$ given by [27]

u_{m} (r, φ, z; t) = \sum_{m} β_{m} (r, z; t) \exp (i m φ),

where the expansion coefficient

β_{m} (r, z, t)

is given by the integral

β_{m} (r, z; t) = \frac{1}{2 π} \int_{0}^{2 π} u_{m} (r, φ, z; t) \exp (- i m φ) d φ .

Proceeding with the ensemble average over atmospheric turbulence for $β_{m} (r, z; t) β_{m}^{*} (r, z; t),$ we have the average intensity of OAM mode $m$ carried by HBP beam

\begin{array}{l} 〈 {| β_{m} (r, z; t) |}^{2} 〉 = {(\frac{1}{2 π})}^{2} \int_{0}^{2 π} \int_{0}^{2 π} u_{m_{0}} (r, φ, z) u_{m_{0}}^{*} (r, φ^{'}, z) \\ \times 〈 \exp [ψ_{x} (r, φ, z) + ψ_{x}^{*} (r, φ^{'}, z) + ψ_{y} (r, φ, z) + ψ_{y}^{*} (r, φ^{'}, z)] 〉 \\ \times \exp [- i m (φ - φ^{'})] 〈 exp [- 2 t^{2} / τ_{0}^{2}] 〉 d φ d φ^{'} . \end{array}

where

〈 \dots 〉

is the ensemble average of atmospheric turbulence, and

〈 \exp [ψ_{x} (r, φ, z)

+ ψ_{x}^{*} (r, φ^{'}, z) + ψ_{y} (r, φ, z) + ψ_{y}^{*} (r, φ^{'}, z)] 〉

is the second-order moment given by [19]

〈 \exp [ψ_{x} (r, φ, z) + ψ_{x}^{*} (r, φ^{'}, z) + ψ_{y} (r, φ, z) + ψ_{y}^{*} (r, φ^{'}, z)] 〉 = \exp [- \frac{1}{2} D (r, φ, φ^{'}; z)],

In Eq. (6), $D (r, φ, φ^{'}; z)$ is the wave structure function of a plane wave given by [19]

D (r, φ, φ^{'}; z) = 4 π^{2} k^{2} z r^{2} [1 - \cos (φ - φ^{'})] \int_{0}^{\infty} κ^{3} ϕ_{n, e f f} (κ) d κ,

with the effective turbulence spectrum of marine atmosphere

ϕ_{n, e f f} (κ)

. The quantity

ϕ_{n, e f f} (κ)

is given by [8]

ϕ_{n, e f f} (κ) = 0.033 C_{n}^{2} κ^{11 / 3} [f (κ l_{0}) g (κ L_{0}) G_{x} (κ) + G_{y} (κ)],

where

κ

is the refractive index fluctuation spatial wavenumber;

C_{n}^{2}

is the refractive index structure constant at the sea surface with units

m^{- 2 / 3}

;

l_{0}

is the inner scale of turbulence;

L_{0}

is the outer scale of turbulence; and

f (κ l_{0})

is defined by [8]

f (κ l_{0}) = \exp (- \frac{κ^{2}}{κ_{H}^{2}}) [1 - 0.061 \frac{κ}{κ_{H}} + 2.836 {(\frac{κ}{κ_{H}})}^{7 / 6}],

where

κ_{H} = 3.41 / l_{0}

.

Outer-scale effects are included by denoting $g (κ L_{0})$ as [28]

g (κ L_{0}) = \frac{κ^{11 / 3}}{{(κ^{2} + κ_{0}^{2})}^{11 / 6}}, κ_{0} = \frac{8 π}{L_{0}} .

Filter functions $G_{x} (κ)$ and $G_{y} (κ)$ are given by [28]

G_{x} (κ) = \exp (- κ^{2} / κ_{x}^{2}),

G_{y} (κ) = \frac{κ^{11 / 3}}{{(κ^{2} + κ_{y}^{2})}^{11 / 6}},

where the quantities

κ_{x}^{2}

and

κ_{y}^{2}

are the spatial-frequency cutoff for the large-scale and small-scale turbulent cell,

κ_{x}^{2} = k η_{x} / z

, and

κ_{y}^{2} = k η_{y} / z

.

In the case of a finite aperture receiver and finite outer-scale, the quantity $η_{x}$ of a plane wave is defined by [9]

η_{x} \approx 2.61 Q_{0} / [2.61 + Q_{0} (1 + 0.65 d^{2} + 0.45 σ_{1}^{2} Q_{H}^{1 / 6})],

where

Q_{H} = 11.628 z / (k l_{0}^{2})

,

σ_{1} = 1.23 C_{n}^{2} k^{7 / 6} z^{11 / 6}

,

Q_{0} = z κ_{0}^{2} / k

, and

d = \sqrt{k D^{2} / (4 z)}

is the circular aperture radius

D / 2

scaled by the Fresnel zone

\sqrt{z / k}

.

Parameter $η_{y}$ is introduced as follows

η_{y} = 3 {(\frac{σ_{1}}{σ_{2}})}^{12 / 5} (1 + 0.69 σ_{2}^{12 / 5}),

where [8]

\begin{array}{l} σ_{2}^{2} = 3.86 σ_{1}^{2} {- \frac{5.581}{Q_{H}^{5 / 6}} + [\sin (\frac{11}{6} \tan^{- 1} Q_{H}) - \frac{0.051 \sin (\frac{4}{3} \tan^{- 1} Q_{H})}{{(1 + Q_{H}^{2})}^{1 / 4}} \\ + \sin (\frac{11}{6} \tan^{- 1} Q_{H}) - \frac{0.051 \sin (\frac{4}{3} \tan^{- 1} Q_{H})}{{(1 + Q_{H}^{2})}^{1 / 4}} + \frac{3.052 \sin (\frac{5}{4} \tan^{- 1} Q_{H})}{{(1 + Q_{H}^{2})}^{7 / 24}}] {(1 + \frac{1}{Q_{H}^{2}})}^{11 / 12}} . \end{array}

The effective marine spectrum of maritime atmosphere in moderate-to-strong scintillation is rewritten as

\begin{array}{l} ϕ_{n, e f f} (κ) = 0.033 C_{n}^{2} [\frac{1}{{(κ^{2} + κ_{0}^{2})}^{11 / 6}} \exp (- \frac{κ^{2}}{κ_{x H}^{2}}) + \frac{1}{{(κ^{2} + κ_{y}^{2})}^{11 / 6}} \\ + \frac{2.836 κ^{7 / 6}}{κ_{H}^{7 / 6} {(κ^{2} + κ_{0}^{2})}^{11 / 6}} \exp (- \frac{κ^{2}}{κ_{x H}^{2}}) - \frac{0.061 κ}{κ_{H} {(κ^{2} + κ_{0}^{2})}^{11 / 6}} \exp (- \frac{κ^{2}}{κ_{x H}^{2}})], \end{array}

where

κ_{x H}^{2} = \frac{κ_{x}^{2} κ_{H}^{2}}{κ_{x}^{2} + κ_{H}^{2}}

.

Substituting Eq. (16) into Eq. (7), we have the wave structure function of a plane wave

\begin{array}{l} D (r, φ, φ^{'}; z) = 0.132 π^{2} k^{2} C_{n}^{2} r^{2} z [1 - \cos (φ - φ^{'})] \int_{0}^{\infty} [\exp (- \frac{κ^{2}}{κ_{x H}^{2}}) \frac{κ^{3}}{{(κ^{2} + κ_{0}^{2})}^{11 / 6}} \\ - \frac{0.061}{κ_{H}} \exp (- \frac{κ^{2}}{κ_{x H}^{2}}) \frac{κ^{4}}{{(κ^{2} + κ_{0}^{2})}^{11 / 6}} + \frac{κ^{3}}{{(κ^{2} + κ_{y}^{2})}^{11 / 6}} + \frac{2.836}{κ_{H}^{7 / 6}} \exp (- \frac{κ^{2}}{κ_{x H}^{2}}) \frac{κ^{25 / 6}}{{(κ^{2} + κ_{0}^{2})}^{11 / 6}}] d κ . \end{array}

Making use of integral formulas [29]

\int_{0}^{\infty} κ^{2 μ} \frac{\exp (- κ^{2} / κ_{H}^{2})}{{(κ^{2} + κ_{0}^{2})}^{11 / 6}} d κ = \frac{1}{2} κ_{0}^{2 μ - 8 / 3} Γ (μ + \frac{1}{2}) U (μ + \frac{1}{2}; μ - \frac{1}{3}; \frac{κ_{0}^{2}}{κ_{H}^{2}}), μ > - \frac{1}{2},

and

\int_{0}^{x} \frac{t^{μ - 1}}{{(1 + β t)}^{ν}} d t = \frac{x^{μ}}{μ} {}_{2}F_{1} (ν, μ; 1 + μ; - β x), μ > 0,

we obtain

\begin{array}{l} \frac{1}{2} D (r, φ, φ^{'}; z) = 2 r^{2} π^{2} k^{2} C_{n}^{2} z [1 - \cos (φ - φ^{'})] \\ \times [0.0165 κ_{0}^{1 / 3} U (2; \frac{7}{6}; \frac{κ_{0}^{2}}{κ_{x H}^{2}}) - \frac{0.013 κ_{0}^{4 / 3}}{κ_{H}} U (\frac{5}{2}; \frac{5}{3}; \frac{κ_{0}^{2}}{κ_{x H}^{2}}) \\ + \frac{0.0661 κ_{0}^{3 / 2}}{κ_{H}^{7 / 6}} U (\frac{31}{12}; \frac{7}{4}; \frac{κ_{0}^{2}}{κ_{x H}^{2}}) + \frac{0.0083 κ_{H}^{4}}{κ_{y}^{11 / 3}} {}_{2}F_{1} (\frac{11}{6}, 2; 3; - \frac{κ_{H}^{2}}{κ_{y}^{2}})], \end{array}

where

U (a; b; z)

is the confluent hypergeometric function of the second kind and

{}_{2}F_{1} (ν, μ; 1 + μ; - β x)

is the hypergeometric function.

From Eq. (20), we obtain the spatial coherence radius in moderate-to-strong scintillation

\begin{array}{l} ρ_{0}^{- 2} = π^{2} k^{2} z C_{n}^{2} [0.0165 κ_{0}^{1 / 3} U (2; \frac{7}{6}; \frac{κ_{0}^{2}}{κ_{x H}^{2}}) - \frac{0.013 κ_{0}^{4 / 3}}{κ_{H}} U (\frac{5}{2}; \frac{5}{3}; \frac{κ_{0}^{2}}{κ_{x H}^{2}}) \\ + \frac{0.0661 κ_{0}^{3 / 2}}{κ_{H}^{7 / 6}} U (\frac{31}{12}; \frac{7}{4}; \frac{κ_{0}^{2}}{κ_{x H}^{2}}) + \frac{0.0083 κ_{H}^{4}}{κ_{y}^{11 / 3}} {}_{2}F_{1} (\frac{11}{6}, 2; 3; - \frac{κ_{H}^{2}}{κ_{y}^{2}})], \end{array}

The average intensity of OAM mode $m$ carried by HBP beam at $r$ in the receiving plane is given by

\begin{array}{l} P_{m} = 〈 {| β_{m} (r, z; t) |}^{2} 〉 \\ = \frac{π}{2 k z} {(m_{0}! A_{0})}^{2} 〈 exp [- 2 \frac{t^{2}}{τ_{0}^{2}}] 〉 \int_{0}^{D / 2} r^{2}^{m_{0} + 1} {| J_{m_{0} / 2} [\frac{k r^{2}}{4 z}] |}^{2} \exp [- \frac{2 r^{2}}{ρ_{0}^{2}}] I_{m - m_{0}} (\frac{2 r^{2}}{ρ_{0}^{2}}) d r, \end{array}

where

D

is the diameter of receiving aperture.

The temporal averaging of a pulse in moderate-to-strong scintillation also can be estimated using the modified Rytov approximation [3]

〈 exp [- 2 \frac{t^{2}}{τ_{0}^{2}}] 〉 = \frac{τ_{0}}{τ_{1}} exp [- 2 \frac{{(t - z / c)}^{2}}{τ_{1}^{2}}],

where

τ_{1} = \sqrt{τ_{0}^{2} + 8 〈 δ τ^{2} 〉}

is the half-width of the received pulse defined by the 1/e point and is called the pulse duration, c is the light speed in atmosphere and

〈 δ τ^{2} 〉

is the turbulent broadening and is given by [2,3]

〈 δ τ^{2} 〉 = \frac{2 π^{2} z}{c^{2}} \int_{0}^{\infty} κ ϕ_{n, e f f} (κ) d κ .

Using Eq. (16), Eq. (24) is rewritten as

\begin{array}{l} 〈 δ τ^{2} 〉 = \frac{0.651 C_{n}^{2} z}{c^{2}} \int_{0}^{\infty} [\exp (- \frac{κ^{2}}{κ_{x H}^{2}}) \frac{κ}{{(κ^{2} + κ_{0}^{2})}^{11 / 6}} + \frac{κ}{{(κ^{2} + κ_{y}^{2})}^{11 / 6}} \\ - \frac{0.061}{κ_{H}} \exp (- \frac{κ^{2}}{κ_{x H}^{2}}) \frac{κ^{2}}{{(κ^{2} + κ_{0}^{2})}^{11 / 6}} + \frac{2.836}{κ_{H}^{7 / 6}} \exp (- \frac{κ^{2}}{κ_{x H}^{2}}) \frac{κ^{13 / 6}}{{(κ^{2} + κ_{0}^{2})}^{11 / 6}}] d κ . \end{array}

Making use of Eq. (18) and Eq. (19), we find that

\begin{array}{l} 〈 δ τ^{2} 〉 = \frac{0.651 C_{n}^{2} z}{c^{2}} {0.5 κ_{0}^{- \frac{5}{3}} U (1; \frac{2}{3}; \frac{κ_{0}^{2}}{κ_{x H}^{2}}) - \frac{0.027}{κ_{H}} κ_{0}^{- \frac{2}{3}} U (\frac{3}{2}; \frac{2}{3}; \frac{κ_{0}^{2}}{κ_{x H}^{2}}) \\ + \frac{1.2645}{κ_{H}^{7 / 6}} κ_{0}^{- \frac{1}{2}} U (\frac{19}{12}; \frac{3}{4}; \frac{κ_{0}^{2}}{κ_{x H}^{2}}) + \frac{κ_{H}^{2}}{2 κ_{y}^{11 / 3}} {}_{2}F_{1} (\frac{11}{6}, 1; 2; - \frac{κ_{H}^{2}}{κ_{y}^{2}})} . \end{array}

Substituting Eq. (26) and Eq. (23) into Eq. (22), we obtain the average intensity of OAM mode $m$ carried by HBP beam

P_{m} (t, z) = \frac{π τ_{0} {(m_{0}! A_{0})}^{2}}{4 k z τ_{1} \sum_{m^{'} = - \infty}^{\infty} p_{m^{'}}} \int_{0}^{D^{2} / 4} x^{m_{0}} {| J_{m_{0} / 2} [\frac{k x}{4 z}] |}^{2} I_{m - m_{0}} (\frac{2 x}{ρ_{0}^{2}}) \exp [- \frac{2 x}{ρ_{0}^{2}} - 2 \frac{{(t - z / c)}^{2}}{τ_{1}^{2}}] d x .

In Eq. (27), we have done the substitution $r = x^{2} .$ For $m = m_{0}$ , the average intensity $P_{m = m_{0}}$ denotes the received energy of the signal OAM mode $m_{0}$ by receiver with diameter $D$ . But for $m \neq m_{0}$ , the energy intensity $P_{m \neq m_{0}}$ denotes the received energy of the crosstalk OAM mode by receiver diameter $D$ .

3. Numerical results and discussions

In this section, we numerically analyze the pulse spreading and average intensity of OAM mode of HBP beam as a function of the refractive index structure constant, wavelength, the outer-scale and inner-scale of turbulence. In the process of numerical computation, the constant which characterizes the beam power is $A_{0} = 10$ , and the input pulse half-width $τ_{0}$ is $5 fs .$ In practical applications, the range of effective pulse duration ( $z / c - τ_{1}$ to $z / c + τ_{1}$ ) of interest is within bounds from $z / c - 1.6 \times 10^{- 14} s$ to $z / c + 16 \times 10^{- 14} s$ [3,9]. Therefore, in our numerical simulation, we set the pulse duration in this range.

In Fig. 1, we plot the pulse width of HBP as a function of $t - z / c$ for 1, 2, 3 and 4 quantum numbers $m_{0}$ of the OAM signal mode, respectively. Other parameter values for the numerical calculation are $L_{0} = 1 m,$ $z = 1 km,$ $C_{n}^{2} = 10^{- 14}$ and $l_{0} = 0.02 m .$ From Fig. 1 we see that the pulse width of OAM signal mode has imperceptible changes under different $m_{0}$ . This result suggests that the intensity transmission of OAM modes with different quantum numbers is almost identical. This phenomenon reveals the feasibility to realize OAM multiplexing. However, the average intensity of the signal OAM mode decreases with the increasing of OAM quantum number, and this result is in keeping with conclusions from other beams carrying OAM modes [11, 20].

Fig. 1 Pulse width of HBP as a function of wavelength for different quantum numbers of OAM signal.

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The pulse width of HBP is shown in Fig. 2 as a function of wavelength for 0, 1, and 2 quantum number differences $Δ m = | m - m_{0} |$ of OAM respectively. Other parameter values for numerical calculation are $L_{0} = 1 m,$ _{$z = 1 km,$} $m_{0} = 1,$ $C_{n}^{2} = 10^{- 14}$ and $l_{0} = 0.02 m .$ Fig. 2 illustrates that, as the beam wavelength increases from 500 nm to 2000 nm, the average intensity of the signal OAM mode increases, and the broadening of pulse is small. This result is because a propagating beam with longer wavelength will experience smaller intensity fluctuation and smaller turbulent expansion in beam cross section [9]. In addition, it is clear that signal OAM mode with longer wavelengths will have a higher signal noise ratio.

Fig. 2 Pulse width of HBP as a function of wavelength for different quantum numbers of OAM.

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In Fig. 3, we draw the pulse width of HBP as a function of turbulent inner-scale for 0, 1, and 2 quantum number differences $Δ m$ of OAM respectively. Other parameters values for numerical calculation are $L_{0} = 1 m,$ _{$z = 1 km,$} $m_{0} = 1,$ _{$C_{n}^{2} = 10^{- 14}$} and $λ = 1550 nm .$ Fig. 3 shows that the pulse width of the signal OAM mode remains essentially unchanged as the inner scale $l_{0}$ increases. From Fig. 3 we also find that, as the inner scale $l_{0}$ increases, the average intensity of signal OAM mode increases, while the intensity of crosstalk OAM modes decrease for $l_{0} > 0.4 mm .$ This is due to the smaller inner scale resulting in stronger scattering disturbance to OAM mode. Consequently, turbulence with smaller inner scale will cause larger spread of energy to adjacent modes. As a result, the mode will have large energy loss from this energy spreading [30].

Fig. 3 Pulse width of HBP as a function of turbulent inner-scale for different quantum number differences of OAM.

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The effects of turbulence outer scale on the average intensity of crosstalk and signal OAM mode can be seen from Fig. 4 for 0, 1, and 2 quantum number differences of OAM respectively. Other parameters values for numerical calculation are $z = 1 km,$ $m_{0} = 1,$ _{$C_{n}^{2} = 10^{- 14},$} $l_{0} = 0.02 m$ and $λ = 1550 nm .$ Fig. 4 reveals that the average intensity of signal and crosstalk OAM modes all decrease with the increasing outer scale from 0.1m to 10m. This is because that the larger outer scale generates stronger beam wander, and stronger beam wander causes a larger deviation of beam center (i.e., intensity maximums) from the receiver center [9]. Therefore, turbulence with larger outer scale will cause a significant decrease in average intensity in both crosstalk and signal OAM modes. Figure 4 also shows that the pulse width of the signal OAM mode remains essentially unchanged as the outer scale increases.

Fig. 4 Pulse width of HBP as a function of turbulent outer-scale for different quantum number differences of OAM.

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The average intensity of crosstalk and signal OAM modes as a function of the refractive index structure constant are shown in Fig. 5 for $L_{0} = 1 m,$ $z = 1 km,$ $m_{0} = 1,$ $λ = 1550 nm$ and $l_{0} = 0.02 m$ . Figure 5(a) shows that as $C_{n}^{2}$ increases, the average intensity of the signal OAM mode decreases, but the pulse width increases. This is caused by stronger turbulent disturbance (larger refractive-index structure parameter) of OAM pulse modes, which not only causes the large spread of energy to adjacent states and energy loss from this spreading, but also leads to more intense scattering and beam spreading from beam scattering. As a consequence, a channel with a larger refractive-index structure parameter, will have lower average intensity of the signal OAM mode and wider pulse width. Correspond to the evolution of the signal OAM mode, the average intensity of crosstalk OAM modes decreases, and the pulse width also increases.

Fig. 5 Pulse width of HBP as a function of the refractive index structure constant of turbulence from $10^{- 15}$ to $10^{- 12}$ for different quantum number differences of OAM.

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4. Conclusion

In summary, we develop a novel statistical model of the average intensity of the OAM mode for the HBP beam in moderate-to-strong scintillation of marine-atmosphere. Our research indicates that the pulse widths of the received OAM signal modes is imperceptible changes for different m₀ and the pulse width of the OAM signal modes is also approximately constant for different outer scales and inner scales of turbulence. The energy loss of the OAM signal mode increases with the increment of the refractive index structure constant and outer-scale of turbulence, but it decreases with the increase of turbulence inner-scale, quantum number of the OAM and wavelength of light.

Funding

Fundamental Research Funds for the Central Universities (Grant No.JUSRP51716A); Postgraduate Research & Practice Innovation Program of Jiangsu Provence (Grant No. KYCX17_1456).

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Effects of moderate to strong turbulence on the Hankel-Bessel-Gaussian pulse beam with orbital angular momentum in the marine-atmosphere

Abstract

1. Introduction

2. Average intensity of OAM modes

3. Numerical results and discussions

4. Conclusion

Funding

References and links

Cited By

Figures (5)

Equations (27)

Optics Express