Turbulent effects of strong irradiance fluctuations on the orbital angular momentum mode of fractional Bessel Gauss beams

Jie Gao; Yixin Zhang; Weiyi Dan; Zhengda Hu

doi:10.1364/OE.23.017024

1. Introduction

The progress in optical quantum communication over the last few years have enabled the development of the orbital angular momentum (OAM) of stable vortex beams whose OAM per photon can take an arbitrary value within a continuous range, either integer or non-integer in units of $ℏ$ [1–7]. It was demonstrated theoretically that the fractional topological charges associated with center vortices and instantons are necessarily grouped into integral charges on a global scale, but are uncorrelated with each other on any shorter scale [7]. Several experiments have verified that light beam may carry fractional OAM when the beam propagates in free-space [8–13]. Such as, S. H. Tao et al. emphasize that a higher-order fractional Bessel beam will be ideal for investigating the behavior of orbital angular momentum at the opening of the fractional Bessel beam [11]. W. M. Lee et al. examine the evolution of optical beams with a fractional phase step hosted within a Gaussian beam by experimental analysis of both the phase and intensity distribution [12]. S. Vyas et al. experimentally demonstrate the generation of a fractional topological charge beam on the basics of vortex lens [13]. This type of light beam has an additional line singularity from beam center to its outward, where the intensity also vanishes.

The fractional Bessel beams (nondiffracting vortex beams) are of special interest due to their properties of divergence-less propagation and self-repair, and have generated widespread interest in the last decade for optical communication [14–17]. J. B. GÖTTE proposed a quantum mechanical description of the beams with fractional topological charges [14] and a high-order Bessel non-vortex beam of fractional type $α$ (HOBNVBs-Fα), was introduced by F. G. Mitri [16,17]. It was found the light modes with fractional orbital angular momentum can be applied to the field of where range from quantum entanglement [14] to optical manipulation [17]. However few researchers have addressed the turbulence effects of strong irradiance fluctuations on OAM modes of fractional Bessel Gauss (FBG) beams.

To understand the turbulence effects of strong irradiance fluctuations on the receiving signals at each point in the receiving plane and the receiving value of acceptor, for FBG beams with orbital angular momentum $l_{0} ℏ$ . In this paper, we outline the influences of atmospheric turbulence on the orbital angular momentum mode probability densities and the normalized powers for FBG beams along the direction of the beam radius. In particular, we draw attention to the models of mode probability density for signal or the crosstalk OAM modes and the normalized power of OAM per photon per unit length of a transverse slice of the beam of a FBG beams in strong turbulence are established. The effects of the fractional order of OAM modes, turbulence strength (in strong fluctuation region), propagation distance and wavelength on the mode probability densities of the signal and crosstalk OAM modes of FBG beams along the direction of $r$ in receiving plane are researched in section 3. Conclusions are presented in Section 4. Our results provide us with new insight into the behavior of the OAM of fractional Bessel Gauss beams in the context of fractional beams and elucidate the connection between traditional optical communication and quantum theory to represent beams with fractional OAM.

2. Ensemble averaging mode probability density

In the strong irradiance fluctuation region [18] and in the half-space $z > 0$ , the normalized complex amplitude of FBG beams in a cylindrical coordinate system $(r, φ, z)$ can be expressed as

E (r, φ, z) = E_{f r e e}^{} (r, φ, z) \exp [ψ_{x} (r, φ, z) + ψ_{y} (r, φ, z)],

where

r = | r |

,

r = (x, y)

is the two-dimensional position vector in the source plane;

z

is propagation distance;

ψ_{x} (r, φ, z)

and

ψ_{y} (r, φ, z)

are the complex phase perturbations due to large-scale and small-scale turbulence eddies, respectively;

E_{f r e e}^{} (r, φ, z)

is the normalized FBG beam at the

z

plane in turbulence-free channel and, in the paraxial channel,

E_{f r e e}^{} (r, φ, z)

has the form [9,19]

E_{f r e e}^{} (r, φ, z) = \frac{1}{μ (z)} \exp [- i \frac{k_{r}^{2} z}{2 k μ (z)} - \frac{r^{2}}{μ (z) w_{0}^{2}}] \sum_{l_{0} = - \infty}^{\infty} C_{γ} J_{| l_{0} |} [\frac{k_{r} r}{μ (z)}] \exp (i l_{0} φ),

where

μ (z) = 1 + i z / z_{R}

,

z_{R} = k w_{0}^{2} / 2

is the Rayleigh range,

k = 2 π / λ

is the wavenumber,

λ

is the wavelength, and

w_{0}

is the beam waist;

J_{| l_{0} |}

is the Bessel function of integer orders,

l_{0}

describes the helical structure of the wave front around a wave front singularity;

k_{r} = ξ k

is the transverse wavenumber;

C_{γ} = \frac{{(\pm i)}^{γ} \sin (\pm π γ)}{π} \cdot \frac{i^{| l_{0} |}}{\pm γ - l_{0}}

,

γ

is any real (integer or fractional) number [9,19]. The diffractive spread of beams propagation in turbulence-free channel from

z

= 0 to 1000m is given by Fig. 1.

Fig. 1 The figure shows beam intensity simulations for fractional FBG beam along the direction $z$ to describe how the beam spreads when the FBG beam transmits in Free Space Optics (FSO). For short distance transmission ( $z$ <200m), the beam is approximate to non-diffracting; for long distance transmission ( $z$ >200m), the diffractive spread of beams become obvious (for example, as $γ$ = 5.1, when $z$ increases from 300m to 1000m, the first ring radius of the fractional FBG beam spreads from 0.13m to 0.48m).

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In a quantum description beams with fractional OAM are superpositions of states of different OAM [10], the FBG is expanded as an summation with basis $\exp (i l φ)$ which carries OAM of $l ℏ$ , then the complex amplitude $E (r, φ, z)$ for FBG can be written as [6,20]

E (r, φ, z) = \sum_{l = - \infty}^{\infty} β_{l} (r, z) \exp (i l φ),

where

β_{l} (r, z)

is the mode amplitude of the mode

l

at the position

(r, z)

, and is given by

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

In quantum description, ${| β_{l} (r, z) |}^{2}$ is the mode probability density of the mode $l$ at the position $(r, z)$ . In turbulent media, the mode probability density is associated with the ensemble average over turbulence medium, i.e. ${〈 {| β_{l} (r, z) |}^{2} 〉}_{a t}$ , where ${〈 \cdot 〉}_{a t}$ represents the ensemble average of the atmospheric turbulence. By Eqs. (1) and (4) and proceeding the ensemble average of the atmospheric turbulence for $β_{l} (r, z) β_{l}^{*} (r, z)$ , and the $*$ denotes complex conjugate. In paraxial channel( $k_{r} \to 0$ ), the ensemble averaging mode probability density ${〈 {| β_{l} (r, z) |}^{2} 〉}_{a t}$ of OAM models $l$ of FBG beam, which is marked with $P_{l} (r, z)$ , is given by

\begin{array}{l} P_{l} (r, z) = {〈 {| β_{l} (r, z) |}^{2} 〉}_{a t} \\ = {(\frac{1}{2 π})}^{2} \int_{0}^{2 π} \int_{0}^{2 π} E_{f r e e}^{} (r, φ, z) E_{f r e e}^{*} (r^{'}, φ^{'}, z) \exp [- i l (φ - φ^{'})] \\ \times {〈 \exp [ψ_{x} (r, φ, z) + ψ_{y} (r, φ, z) + ψ_{x}^{*} (r^{'}, φ^{'}, z) + ψ_{y}^{*} (r^{'}, φ^{'}, z)] 〉}_{a t} d φ d φ^{'}, \end{array}

where

{〈 \exp [ψ_{x} (r, φ, z) + ψ_{y} (r, φ, z) + ψ_{x}^{*} (r^{'}, φ^{'}, z) + ψ_{y}^{*} (r^{'}, φ^{'}, z)] 〉}_{a t}

is given by

\begin{array}{l} {〈 \exp [ψ_{x} (r, φ, z) + ψ_{y} (r, φ, z) + ψ_{x}^{*} (r^{'}, φ^{'}, z) + ψ_{y}^{*} (r^{'}, φ^{'}, z)] 〉}_{a t} \\ = \exp [- \frac{1}{2} D (r, φ, r^{'}, φ^{'}; z)], \end{array}

and

D (r, φ, r^{'}, φ^{'}; z) = D_{x} (r, φ, r^{'}, φ^{'}; z) + D_{y} (r, φ, r^{'}, φ^{'}; z)

. The

D_{x} (r, φ, r^{'}, φ^{'}; z)

and

D_{y} (r, φ, r^{'}, φ^{'}; z)

are wave structure functions caused by large-scale and small-scale turbulence eddies, respectively [21].

Based on the modified Rytov method [18,21], the effective atmospheric spectral model is given by

Φ_{n} = 0.033 C_{n}^{2} κ^{- 11 / 3} [f (κ l_{i}) g (κ L_{o}) G_{x} + G_{y}],

where

κ

is the spatial wave number of refractive index fluctuations,

C_{n}^{2}

is the generalized refractive-index structure parameter with units m^-2/3;

f (κ l_{i}) = \exp (- κ^{2} / κ_{m}^{2})

,

g (κ L_{o}) = {(κ^{2} + κ_{o}^{2})}^{- 11 / 6}

,

κ_{m} = 5.92 / l_{i}

,

l_{i}

denotes the inner scalar of turbulence and

κ_{0} = 2 π / L_{o}

,

L_{o}

denotes the outer scaler of turbulence;

G_{x} (κ)

and

G_{y} (κ)

represent the large and small scale filter functions, respectively. In far field region and strong irradiance fluctuation region [21], the filter function are given by

G_{x} (κ) = \exp [- κ^{2} / κ_{x}^{2}], G_{y} (κ) = \frac{κ^{11 / 3}}{{(κ^{2} + κ_{x}^{2})}^{11 / 6}},

where

κ_{x}^{2} = k η_{x} / z

is the large scale frequency cut-off and

κ_{y}^{2} = k η_{y} / z

is the small scale frequency cut-off,

η_{x} = \frac{8.56}{1 + 0.08 σ_{R}^{2} {(10.89 z / k l_{i}^{2})}^{1 / 6}}

,

σ_{R}^{2}

is the Rytov variance which indicates the strength of strong irradiance fluctuations, and given by

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

,

η_{y} = 9 [1 + 0.69 {(0.4 σ_{R}^{2})}^{6 / 5}]

[22].

On spherical wave approximation and by the approximation of generalized hypergeometric functions in [21], we have the large and small scale portions of the wave structure function for a spherical wave

\begin{array}{l} D_{x} (r, r^{'}, φ, φ^{'}; z) = 0.604 C_{n}^{2} k^{2} z κ_{x m}^{1 / 3} {[r^{2} + {r^{'}}^{2} - 2 r r^{'} \cos (φ - φ^{'})] \\ \times {[1 + 0.033 κ_{x m}^{2} [r^{2} + {r^{'}}^{2} - 2 r r^{'} \cos (φ - φ^{'})]]}^{- 1 / 6}}, \end{array}

\begin{array}{l} D_{y} (r, r^{'}, φ, φ^{'}; z) = 0.604 C_{n}^{2} k^{2} z κ_{m}^{1 / 3} {[r^{2} + {r^{'}}^{2} - 2 r r^{'} \cos (φ - φ^{'})] \\ \times {[1 + 0.033 κ_{m}^{2} [r^{2} + {r^{'}}^{2} - 2 r r^{'} \cos (φ - φ^{'})]]}^{- 1 / 6}} \\ - 0.604 C_{n}^{2} k^{2} z κ_{y m}^{1 / 3} {[r^{2} + {r^{'}}^{2} - 2 r r^{'} \cos (φ - φ^{'})] \\ \times {[1 + 0.033 κ_{y m}^{2} [r^{2} + {r^{'}}^{2} - 2 r r^{'} \cos (φ - φ^{'})]]}^{- 1 / 6}}, \end{array}

where

κ_{x m}^{2} = \frac{35.046 k η_{x}}{k η_{x} l_{i}^{2} + 35.046 z}

and

κ_{y m}^{2} = \frac{35.046 k η_{y}}{k η_{y} l_{i}^{2} + 35.046 z}

.

For strong irradiance fluctuations $(σ_{R}^{2} ≫ 1)$ and by approximation ${(\frac{k}{z} [r^{2} + {r^{'}}^{2} - 2 r r^{'} \cos (φ - φ^{'})])}^{5 / 6}$ $\approx (\frac{k}{z} [r^{2} + {r^{'}}^{2} - 2 r r^{'} \cos (φ - φ^{'})])$ [12], we have

D_{x} (r, r^{'}, φ, φ^{'}; z) \approx 0.604 C_{n}^{2} k^{2} z κ_{x m}^{1 / 3} [r^{2} + {r^{'}}^{2} - 2 r r^{'} \cos (φ - φ^{'})],

\begin{array}{l} D_{y} (r, r^{'}, φ, φ^{'}; z) \approx 0.867 σ_{R}^{2} \frac{k}{z} [r^{2} + {r^{'}}^{2} - 2 r r^{'} \cos (φ - φ^{'})] \\ - 0.604 C_{n}^{2} k^{2} z κ_{y m}^{1 / 3} [r^{2} + {r^{'}}^{2} - 2 r r^{'} \cos (φ - φ^{'})] . \end{array}

On substituting the Eqs. (11) and (12) into Eq. (6), we obtain

\begin{array}{l} \exp [- \frac{1}{2} D (r, φ, r^{'}, φ^{'}; z)] = \exp {- [0.604 C_{n}^{2} k^{2} z κ_{x m}^{1 / 3} + (0.867 σ_{R}^{2} \frac{k}{z} - 0.604 C_{n}^{2} k^{2} z κ_{y m}^{1 / 3})] \\ \times [r^{2} + {r^{'}}^{2} - 2 r r^{'} \cos (φ - φ^{'})]} \\ = \exp [- \frac{r^{2} + {r^{'}}^{2} - 2 r r^{'} \cos (φ - φ^{'})}{ρ_{0}^{2}}], \end{array}

where is the new spatial coherence radius of a spherical wave propagating in the strong irradiance fluctuation channel and is given by

ρ_{0} = {0.491 σ_{R}^{2} \frac{k}{z} {{(\frac{35.046 η_{x} z}{k η_{x} l_{i}^{2} + 35.046 z})}^{1 / 6} + [1.766 - {(\frac{35.046 η_{y} z}{k η_{y} l_{i}^{2} + 35.046 z})}^{1 / 6}]}}^{- 1 / 2} .

Based on the integral expression [23]

\int_{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),

Where

w (z) = w_{0} \sqrt{1 + z^{2} / z_{R}^{2}}

is the Gaussian spot size;

I_{n} (η)

is the Bessel function of second kind with

n

order, with the help of Eqs. (2), (5) and (6), we have the mode probability density of the OAM for FBG beams in paraxial turbulence channel

P_{l} (r, z) = \frac{w_{0}^{2}}{w^{2} (z)} \sum_{l_{0} = - \infty}^{\infty} C_{γ}^{2} {| J_{| l_{0} |} (\frac{k_{r} r}{μ (z)}) |}^{2} \exp [- (\frac{1}{w^{2} (z)} + \frac{1}{ρ_{0}^{2}}) 2 r^{2}] I_{l - l_{0}} (2 r^{2} / ρ_{0}^{2}),

where

Δ l = l - l_{0}

denotes the value of crosstalk; for

Δ l = 0

,

P_{l_{0}} (r, z)

is the mode probability density of the signal OAM mode

l_{0}

for FBG beams, and for

Δ l \neq 0

,

P_{l} (r, z)

is the mode probability density of the crosstalk OAM mode, which represents the probability density of the part of the energy launched into the signal OAM mode redistributed into other OAM modes by atmospheric turbulence [24].

The normalized power of OAM per photon per unit length of a transverse slice of the beam is calculated with the following expression [9]

L_{z} (l) = ℏ \sum_{l = - \infty}^{\infty} \frac{l \cdot B P_{l}}{\sum_{l = - \infty}^{\infty} B P_{l}} .

where

\sum_{l = - \infty}^{\infty} B P_{l}

is the beam power with

B P_{l} = \frac{w_{0}^{2}}{w^{2} (z)} \sum_{l_{0} = - \infty}^{\infty} C_{γ}^{2} \int_{0}^{\infty} {| J_{| l_{0} |} (\frac{k_{r} r}{μ (z)}) |}^{2} \exp [- (\frac{1}{w^{2} (z)} - \frac{1}{ρ_{0}^{2}}) 2 r^{2}] I_{l - l_{0}} (2 r^{2} / ρ_{0}^{2}) r d r,

and

B P_{l}

is the power of each constituent angular harmonics beam, which is given by [9]. Similarly, for

l = l_{0}

,

L_{z} (l_{0})

expresses the normalized power of the signal OAM mode for FBG beams, and for

l \neq l_{0}

,

L_{z} (l)

is the normalized powers of the crosstalk OAM modes.

3. Numerical discussion

To investigate the influences of the strong irradiance fluctuations fractional order of OAM modes $γ$ , the deviation of quantum number $Δ l$ , the Rytov variance $σ_{R}^{2}$ and beam waists $w_{0}$ on the average probability density of signal OAM modes and signal normalized powers, in this section, we employ the method of numerical calculation to analyze the Eqs. (16) and (17). Obviously, the average probability density of signal OAM modes and signal normalized powers decrease, and the average crosstalk probability of OAM modes and crosstalk normalized powers increase when the propagation distance $z$ increases. Then we do not discuss the influences owed to the propagation distance $z$ . To simplify, in figures, we write ensemble average probability density ( $P_{l_{0}} (r, z)$ ) of the signal OAM modes to mode probability, and write ensemble average probability density ( $P_{l \neq l_{0}} (r, z)$ ) of the crosstalk OAM modes to crosstalk probability.

In this section, we first study the effects of $γ$ , $Δ l$ , $σ_{R}^{2}$ and $w_{0}$ on the mode probability densities of the signal ( $P_{l_{0}} (r, z)$ ) and crosstalk ( $P_{l \neq l_{0}} (r, z)$ ) OAM modes for FBG beams along the direction of beam radius $r$ in receiving plane. The simulation results are shown in Figs. 2-5. The parameters are taken as $λ$ = 1550nm, $z$ = 1km, $w_{0}$ = 0.1m, $γ$ = 5.5, $ξ$ = 0.001, $α$ = 11/3, $l_{i}$ = 0.001m, $σ_{R}^{2}$ = 10, $Δ l$ = 0, 1, 2 and 3, unless otherwise indicated. Finally, we evaluate the effects of the irradiance fluctuations on the normalized powers $L_{z} (l)$ in parameter $Δ l = 0$ or $Δ l = 1$ . The simulation results are shown in the final figure.

Figure 2 plots the performance of $P_{l} (r, z)$ of FBG beam versus $r$ from 0 to 0.2m for the different $Δ l$ . Figure 2 demonstrates that the $P_{l_{0}} (r, z)$ decays rapidly and tends to a stable value when r increases. As can be seen from Fig. 2, the peak of $P_{l \neq l_{0}} (r, z)$ moves outward and downward clearly as $Δ l$ gets larger. When $r$ is large enough (in Fig. 2, $r$ >0.05m), $P_{l_{0}} (r, z)$ and $P_{l \neq l_{0}} (r, z)$ tend to be same. The results show that we can obtain the higher signal-noise ratio (SNR) only in the circle region of radius $r$ <0.05m, and the center of the circle region is beam center. So in the next context, we adopt the circle region of radius $r$ <0.05m.

Fig. 2 The average mode probability $P_{l} (r, z)$ of FBG beam along the direction of the beam radius $r$ for different values of the quantum number deviation $Δ l $ .

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In Fig. 3, $P_{l} (r, z)$ is plotted as a function of $r$ for mode crosstalk $Δ l$ = 0 and 1. Figures 3 (a) and 3(b) is for $γ$ = 5.1, 5.3, 5.5, 5.7 and 5.9, respectively. As is seen from Fig. 3, $P_{l} (r, z)$ are the vibration curves for $γ$ = 5.1, 5.3, 5.7 and 5.9, and these vibration curves are around the curve of $γ$ = 5.5. Figure 3(a) indicates that $P_{l_{0}} (r, z)$ decrease with the increasing of $r$ , while $P_{l \neq l_{0}} (r, z)$ decay after the increase in the region close to circle area to circle area in Fig. 3(b). It is interesting to note that when the value of $γ$ is half integer, its curve is without vibration in Figs. 3(c) and 3(d). Obviously, the closer the value of $γ$ to half integer, the closer its curve tends to the image of $γ$ = 5.5. In order to facilitate the beam properties, we set $γ$ = 5.5.

Fig. 3 The average probability densities $P_{l} (r, z)$ of OAM modes for fractional FBG beam along the direction of the beam radius $r$ with different values of $γ$ when (a) $Δ l = 0$ , signal probability densities; (b) $Δ l = 1$ , crosstalk probability densities; (c) $γ$ is half-integer, and $Δ l = 0$ signal probability densities; (d) $γ$ is half-integer, and $Δ l = 1$ crosstalk probability densities.

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Figure 4 indicates the variation of $P_{l} (r, z)$ against $r$ for selected beam waist $w_{0}$ = 0.02m, 0.05m and 0.10m. Figure 4 shows that increasing $w_{0}$ can rise $P_{l} (r, z)$ . From Fig. 4(a), it can be seen that the lower $w_{0}$ is, the faster $P_{l_{0}} (r, z)$ decays. And the signal difference of adjacent channel increases as the detection position away from the beam center. As is shown by Fig. 4(b), all the curves pick up and then fall, and the value of $P_{l \neq l_{0}} (r, z)$ keeps growing as $w_{0}$ gets larger.

Fig. 4 The average probability densities $P_{l} (r, z)$ of OAM modes for fractional FBG beam along the direction of the beam radius $r$ with different values of beam waist $w_{0}$ when (a) $Δ l = 0$ , signal probability densities; (b) $Δ l = 1$ , crosstalk probability densities.

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Figure 5 illustrates that the variation of $P_{l} (r, z)$ along the radial direction $r$ from 0 to 0.05m. For this example we have chosen $σ_{R}^{2}$ = 5, 10 and 30. According to Fig. 5(a), as $σ_{R}^{2}$ increases, $P_{l_{0}} (r, z)$ decreases slightly but not obvious. It’s clear that the influence of turbulence on $P_{l_{0}} (r, z)$ is more apparent as $r$ increases in the strong fluctuation region. Near the central area of the beam, $P_{l \neq l_{0}} (r, z)$ increases with the increasing of $σ_{R}^{2}$ and the main energy still remains at the transmit signal channel. However, as the detection position away from the beam center, the energy of crosstalk channel increases immediately and is almost as strong as the energy that remains in the signal channel. When detection position is far away from the beam center in receiving plane, $P_{l_{0}} (r, z)$ and $P_{l \neq l_{0}} (r, z)$ tend to be same. For the case of $σ_{R}^{2}$ = 5, the energy of crosstalk channel at $r$ = 0.05m is almost the same as the energy of signal channel. The peak of the $P_{l \neq l_{0}} (r, z)$ shifts to the beam center, and half width of the peak narrows down as $σ_{R}^{2}$ increases, i.e. the increase of $σ_{R}^{2}$ can make the crosstalk noise stronger and more concentrated.

Fig. 5 The average probability densities $P_{l} (r, z)$ of OAM modes for fractional FBG beam along the direction of the beam radius $r$ with different values of Rytov variance $σ_{R}^{2}$ when (a) $Δ l = 0$ , signal probability densities; (b) $Δ l = 1$ , crosstalk probability densities.

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Finally, in Fig. 6, we describe the effects of Rytov variance $σ_{R}^{2}$ on the normalized powers $L_{z} (l)$ in the case that the inner scales $l_{i}$ tend to 0. Comparing the curves in Fig. 6, as $σ_{R}^{2}$ increases, the normalized power of the signal OAM mode reduces more and the normalized powers of the crosstalk OAM modes increase more. As is seen from this figure, the normalized powers of the signal or crosstalk OAM modes are the complicated functions of beam waist.

Fig. 6 The normalized powers $L_{z} (l)$ of fractional OAM modes for fractional FBG beam along beam waist $w_{0}$ with different values of turbulence fluctuation $σ_{R}^{2}$ when (a) $Δ l = 0$ , signal normalized powers $L_{z} (l_{0})$ ; (b) $Δ l = 1$ , crosstalk normalized powers $L_{z} (l \neq l_{0})$ .

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4. Conclusions

In conclusion, the model of average probability densities and the normalized powers of the signal/crosstalk OAM modes for FBG beams in the turbulent atmosphere of strong irradiance fluctuations have been developed. This study has revealed that average mode probability $P_{l_{0}} (r, z)$ and average crosstalk probability $P_{l \neq l_{0}} (r, z)$ are the same when beam radius $r$ is large enough. For FBG beams, when $r$ varies, $P_{l_{0}} (r, z)$ and $P_{l \neq l_{0}} (r, z)$ oscillate except half-integer, and the larger the parameter $γ$ is, the greater the amplitude of oscillation curves will be. For beams of $γ$ = 5.5, the average mode probability $P_{l_{0}} (r, z)$ decays with the decreasing of the beam waist $w_{0}$ or the increasing of irradiance fluctuations $σ_{R}^{2}$ along beam radius, and average crosstalk probability $P_{l \neq l_{0}} (r, z)$ drops rapidly after the slow growth as $r$ increases, namely average crosstalk probability curves have peaks. Near the central region of beam, enlarging $Δ l $ or minifying $σ_{R}^{2}$ and $w_{0}$ can make $P_{l \neq l_{0}} (r, z)$ drop down. The normalized powers of signal or crosstalk OAM modes are the complicated functions of beam waist. Furthermore, lower $σ_{R}^{2}$ can more efficient reduce the turbulence effect on the normalized power $L_{z} (l_{0})$ of signal OAM modes, which is opposite to crosstalk normalized powers $L_{z} (l \neq l_{0})$ .

Acknowledgments

This work is supported by the Natural Science Foundation of Jiangsu Province of China (Grant No. BK20140128) and the National Natural Science Foundation of Special Theoretical Physics (Grant No. 11447174).

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Turbulent effects of strong irradiance fluctuations on the orbital angular momentum mode of fractional Bessel Gauss beams

Abstract

1. Introduction

2. Ensemble averaging mode probability density

3. Numerical discussion

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (6)

Equations (18)

Optics Express