Influence of atmospheric turbulence on the transmission of orbital angular momentum for Whittaker-Gaussian laser beams

Yixin Zhang; Mingjian Cheng; Yun Zhu; Jie Gao; Weiyi Dan; Zhengda Hu; Fengsheng Zhao

doi:10.1364/OE.22.022101

1. Introduction

The mode-division multiplexing is one of recently discussed options for space-division multiplexing [1], the multimode wireless turbulence space optical schemes resorting to orbital angular momentum (OAM) were demonstrated in [2–9]. In particular, Laguerre-Gaussian (LG) modes with different radial orders p and angular momentum orders l are good candidates for space-division multiplexing [2–9]. One of the biggest challenges that confronts the use of the OAM modes for quantum communication is the effects of the atmospheric turbulence during transmission over long propagation distance. The atmospheric turbulence aberrations can cause the crosstalk among the OAM modes of single photons, reduce information capacity of the communication channel [2, 4, 5], induce the spread of the spiral spectrum of OAM modes [3, 8], and the attenuation or crosstalk among channels in the free-space optical communication systems [6, 7, 9]. The investigation of the probability density of the OAM modes for Hankel-Bessel (HB) non-diffracted beams in paraxial turbulence channel has revealed that the non-diffracted beams can effectively mitigate the effect of the diffraction in free space and the influence of atmospheric turbulence on the transmission [10]. However, the number of adjustable parameters of HB beams is limited, which is the disadvantage for the optimization design of the system for resistance against turbulence interference. Compared with HB beams, Whittaker-Gaussian (WG) laser beams are the non-diffracted beams with several adjustable parameters. So it is more significant to study the transmission rules of WG beams in turbulent atmosphere. The propagation through complex ABCD optical systems, normalization factor, beamwidth, the quality M² factor and the kurtosis parameter of the WG beams had been studied in [11]. However, to the best of our knowledge, there are almost no discussion with respect to the effects of turbulence on the transmission of the OAM modes for WG beams in slant non-Kolmogorov turbulence channels.

In this paper we model the effects of turbulence on the detection probability spectrum and the mode weight of the OAM mode for WG beams through in a slant non-Kolmogorov turbulence atmosphere.

2. The probability distribution of spiral plane modes

In cylindrical coordinates, the electric field distribution of WG modes at the z plane in a paraxial channel of free atmospheric turbulence can be described as [11, 12]

\begin{matrix} W G_{l_{0}, i γ} (r, φ, z) = C_{l_{0}, i γ} \frac{{(1 - z / q_{0}^{*})}^{i γ / 2 - 1 / 2}}{{(1 + z / q_{0})}^{i γ / 2 + 1 / 2}} \exp (\frac{i k r^{2}}{2 (z + q_{0})}) \\ \times {(P r^{2})}^{l_{0} / 2} {}_{1}F_{1} (β, l_{0} + 1, P r^{2}) \exp (i l_{0} φ) \end{matrix}

where z is the propagation distance of light; q₀ is the beam parameter at the plane z = 0 with

1 / q_{0} = 1 / R + i 2 / k w_{0}^{2}

; R is the radius of curvature of the initial spherical phase front;

q_{0}^{*}

is the complex conjugate of q₀;

k = 2 π / λ

is the wave number of light; λ is the wavelength; w₀ is the beamwidth of the Gaussian envelop; l₀ corresponds to the OAM

l_{0} ℏ

carried by the beam, which describes the helical structure of the wave front around a wave front singularity;

r = | r |, r = (x, y)

is the two-dimensional position vector in the source plane, φ is the azimuthally angle;

{}_{1}F_{1} [a, b, x] = \sum_{n = 0}^{\infty} \frac{{(a)}_{n}}{n! {(b)}_{n}} x^{n}

is the confluent hypergeometric function;

β = (l_{0} + 1 - i γ) / 2

;

C_{l_{0}, i γ}

is the normalization constant of WG beams, which is determined by

C_{l_{0}, i γ} = 1 / \sqrt{σ_{l_{0}, i γ}^{0}}

and

σ_{l_{0}, i γ}^{0}

is given by

σ_{l_{0}, i γ}^{0} = \frac{π l_{0}! w_{0}^{2}}{2^{2 l_{0} + 1}} \frac{ζ^{2 l_{0}}}{| {(1 - i ζ^{2} / 4)}^{1 + l_{0} - i γ} |} \times F (\frac{1 + l_{0} - i γ}{2}, \frac{1 + l_{0} + i γ}{2}; l_{0} + 1; \frac{ζ^{4}}{16 + ζ^{4}})

where W₀ and γ are real parameters of WG beams;

ξ = w_{0} / W_{0}

, and

P = \frac{2}{w_{0}^{2} {(z^{2} + 1) [(1 / R^{2}) + 4 {(k w_{0}^{2})}^{- 2}] + 2 z / R}}

For l₀ = 0 and $ζ ≪ 1$ , the beam transverse intensity behaves as a Gaussian beam, and the beam has a doughnut-like structure for $l_{0} > 0$ [12].

In the weak atmospheric turbulence region [13] and at any point in the half-space z>0, as an approximate, we can express the complex amplitude $W {G^{'}}_{l, i γ} (r, φ, z)$ of WG beams as

W {G^{'}}_{l, i γ} (r, φ, z) = W G_{l_{0}, i γ} (r, φ, z) \exp [ψ_{1} (r, φ, z)]

where

ψ_{1} (r, φ, z)

is complex phase perturbation of spherical waves propagating through turbulence.

Based on the discussion in Ref [14, 15], we can write the function $W {G^{'}}_{l, i γ} (r, φ, z)$ as a superposition of the spiral harmonics $\exp (i l φ)$

W {G^{'}}_{l, i γ} (r, φ, z) = \frac{1}{\sqrt{2 π}} \sum_{l = - \infty}^{\infty} β_{l} (r, z) \exp (- i l φ)

where

β_{l} (r, z)

is given by the integral

β_{l} (r, z) = \frac{1}{\sqrt{2 π}} \int_{0}^{2 π} W {G^{'}}_{l, i γ} (r, φ, z) \exp [i l φ] d φ

Following the same procedure as in Ref [3], the mode probability distribution of the spiral plane mode with phase $\exp (i l φ)$ can be expressed as

{| β_{l} (r, z) |}^{2} = \frac{1}{2 π} \int_{0}^{2 π} \int_{0}^{2 π} W {G^{'}}_{l, i γ} (r, φ, z) W {G^{'}}_{l, i γ}^{*} (r, φ, z) \exp [- i l (φ - φ^{'})] d φ^{'} d φ

By substituting Eq. (4) into Eq. (7) and averaging over turbulence ensembles, we have the ensembles averaging mode probability distribution of the spiral plane waves with phase $\exp (i l φ)$ of WG beams propagating in paraxial turbulence channel, which is named as spiral mode probability distribution and is given by

\begin{matrix} 〈 {| β_{l} (r, z) |}^{2} 〉 = \frac{1}{2 π} \int_{0}^{2 π} \int_{0}^{2 π} W G_{l_{0}, i γ} (r, φ, z) W G_{l_{0}, i γ}^{*} (r, φ, z) \exp [- i l (φ - φ^{'})] d φ^{'} d φ \\ \times \exp [- (2 r^{2} - 2 r^{2} \cos (φ^{'} - φ)) / ρ_{0}^{2}] \end{matrix}

where

〈 \cdot 〉

represent an ensemble average over the turbulence statistics, ρ₀ is the spatial coherence radius of a spherical wave propagating in slant non-Kolmogorov turbulence [16] and is given by

ρ_{0} = {\frac{2 Γ (\frac{3 - α}{2}) {[\frac{8}{α - 2} Γ (\frac{2}{α - 2})]}^{(α - 2) / 2}}{π^{1 / 2} k^{2} Γ (\frac{2 - α}{2}) \int_{0}^{z} C_{n}^{2} (ξ, θ) {(1 - ξ / z)}^{α - 2} d ξ}}^{1 / (α - 2)}, 3 < α < 4

where α is the non-Kolmogorov turbulence-parameter,

Γ (x)

is the Gamma function,

C_{n}^{2} (ξ, θ)

is the refractive index structure parameter in a slant turbulence channel with units

m^{3 - α}

and θ is the zenith angle [17].

By substituting Eq. (1) into Eq. (8) and utilizing the integral expression [18]

\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),

we have the final expression of the spiral mode probability distribution of WG beams

\begin{array}{l} 〈 {| β_{l} (r, z) |}^{2} 〉 = \frac{{(P r^{2})}^{l_{0}}}{σ_{l_{0}, i γ}^{0}} \frac{\exp [- γ (\arg δ + \arg δ^{'})]}{\sqrt{{(1 - z^{2})}^{2} + {(2 z / k w_{0}^{2})}^{2}}} \exp {- \frac{2 r^{2}}{w_{0}^{2} η [{(z + 1 / R η)}^{2} + {(2 / k η w_{0}^{2})}^{2}]}} \\ \times {| {}_{1}F_{1} (β, l_{0} + 1, P r^{2}) |}^{2} I_{l - l_{0}} (\frac{2 r^{2}}{ρ_{0}^{2}}) \end{array}

where

\arg δ = arc \tan [\frac{2 z / k w_{0}^{2}}{1 - z / R}]

,

\arg δ^{'} = arc \tan [\frac{2 z / k w_{0}^{2}}{1 + z / R}]

,

η = {(1 / R)}^{2} + {(2 / k w_{0}^{2})}^{2}

and

I_{n} (η)

is the Bessel function of second kind with n order.

By defining the mode weight (or energy content) for each angular momentum mode m with the signal OAM mode l₀

ℑ_{m} = \int_{0}^{\infty} 〈 {| β_{m} (r, z) |}^{2} 〉 r d r

and using the definition of the detection probability spectrum (energy content or weight of each of the spiral harmonics of any field distribution) in Ref [14, 15], we can calculate the detection probability spectrum of the OAM modes for WG beams at the z plane after propagating in non-Kolmogorov turbulence by following relationship

℘_{l} (z) = ℑ_{l} (z) / \sum_{m = - \infty}^{\infty} ℑ_{m} (z)

For designated OAM mode l = l₀, $℘_{l_{0}} (z)$ (Eq. (13)) is defined as the detection probability of the signal OAM mode l₀, which shows the transfer rate of the transmitted OAM state; for $l = l_{0} \pm Δ l$ , $℘_{l} (z)$ is defined as the crosstalk probability which denotes the probability of a photon to change its OAM state, and ∆l denotes the OAM quantum number difference.

3. Numerical discussion

In this section, we discuss the numerical results of the detection probability spectrum, the detection probability, crosstalk probability and the mode weight of the signal OAM mode for WG beams in non-Kolmogorov turbulence by using the analytical equations in the previous section with the radius of curvature of the initial spherical phase front R = $5 \times 10^{4} m$ and the zenith angle θ = $0^{\circ}$ .

In Figs. 1(a)-1(b), we investigate the impact of source model for WG beams (a) and LG beams (b) (based on Eq. (14) in [3]) on the detection probability of the signal OAM mode. To this end, we fix the parameters of both WG and LG beams as wavelength λ = 1550nm, the beamwidth w₀ = 0.02m, the refractive index structure parameter $C_{n}^{2} = 10^{- 15} m^{3 - α}$ , α = 3.97, the real parameters of WG beams W₀ = $1 \times 10^{- 3}$ , γ = 1, and vary the propagation distance z from 0 to 1000m and the signal OAM mode quantum number l₀ from 1 to 4. Comparing the subplots from Figs. 1(a)-1(b), it can be said that WG beams provide excellent performance improvement on the detection probability of the signal OAM mode for LG beams in the weak turbulent regimes. However, as propagation distance $z \geq 500 m$ , the decrease of the detection probability of the signal OAM mode for WG beams is more noticeable with the increase of z. Figures 1(a)-1(b) also show that increasing quantum number l₀ will lead to a significant effect on the detection probability of the signal OAM mode for both WG and LG beam, and the detection probability decreases with the increase of quantum number l₀ and the propagation distance z. It is because the OAM modes with larger quantum number l₀ will have a larger radius after propagation and will result in an increase in the negative effects of turbulence, therefore it can lead to the decrease of the detection probability of the signal OAM mode.

Fig. 1 Detection probability of the signal OAM mode for WG beams (a) and LG beams (b) against z for l₀

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With the same parameters setting used in Figs. 1(a)-1(b), we evaluate the crosstalk probability with the OAM quantum number difference ∆l = |l-l₀| = 1 and the detection probability spectrum of the signal OAM mode l₀ = 1, 2, 3 and 4 for WG beams in Fig. 2 and Fig. 3. Figure 2 shows that the crosstalk probability increases with the increasing of the signal OAM mode quantum number l₀ and propagation distance z. As shown in Fig. 3, the majority of the crosstalk for the WG modes happens with the immediate neighboring modes (∆l = 1), and the other crosstalk effect on the detection of the signal OAM mode for WG beams can be ignored, from this point, we can said that the resistant ability to turbulence interference of WG beams is much better than LG beams [5, 7]. It is also observed that the detection probability of the signal OAM mode decreases with the increasing of the quantum number l₀.

Fig. 2 Crosstalk probability of the signal OAM mode for WG beams for z against l₀, with ∆l = 1

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Fig. 3 Detection probability spectrum of OAM modes for WG beams against l for l_0.

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Figure 4 explores the impact of the beamwidth w₀ on the detection probability of the signal OAM mode l₀ = 1 for WG beams by changing w₀ = 0.01m, 0.02m and 0.03m in the different turbulence $C_{n}^{2}$ from $10^{- 16}$ to $10^{- 15}$ , under the conditions with λ = 1550nm, α = 3.97, W₀ = $1 \times 10^{- 3}$ , γ = 1 and z = 1000m. We can see that increasing beamwidth w₀ will lead to a significant improvement on the detection of the signal OAM mode for WG beams. This is reasonable, since the decrease of w₀ will result in stronger diffraction and larger mode crosstalk at the receiver. These results show that increasing the beamwidth of the laser beams is a good choice to improve the SNR of the communication system in turbulent atmosphere.

Fig. 4 Detection probability of the signal OAM mode for WG beams against $C_{n}^{2}$ for w_0.

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Assuming parameters as w₀ = 0.02m, $C_{n}^{2} = 10^{- 15} m^{3 - α}$ , γ = 1, W₀ = $1 \times 10^{- 3}$ and z = 1000m, Fig. 5 depicts the detection probability of the signal OAM mode l₀ = 1 for WG beams versus non-Kolmogorov parameter α for different wavelength λ = 1550nm, 850nm and 632.8nm. It is shown that the detection probability increases with the increase of wavelength λ. From Figs. 4 and 5,we also can observe that the increases of the values of $C_{n}^{2}$ and α result in the decrease of the detection probability, and increasing α from 3.07 to 3.97, the detection probability of the signal OAM mode (l₀ = 1) will give rise to decrease only about 10%, and the influence is obvious when α from 3.77 to 3.97. Although the influence of the zenith angle is not displayed here explicitly, the change tendency of the detection probability versus the zenith angle of the slant non-Kolmogorov turbulence channel is analogous to that versus associated $C_{n}^{2}$ in Fig. 4. It is clearly that the smaller $C_{n}^{2}$ and α will cause the larger coherence length $ρ_{0}$ , and the larger $ρ_{0}$ will attribute the improvement in the detection probability to a smaller ratio of the mode crosstalk.

Fig. 5 Detection probability of the signal OAM mode for WG beams against α for λ.

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Figures 6(a)-6(b) exhibit the effect of the real parameter of WG beams γ on the detection probability and the mode weight of the signal OAM mode l₀ = 1 for WG beams by changing γ from 0 to 2 in the different turbulence $C_{n}^{2}$ from $5 \times 10^{- 16}$ , $7.5 \times 10^{- 16}$ to $10^{- 15}$ . Here, we set parameters as λ = 1550nm, w₀ = 0.02m, α = 3.97, W₀ = $1 \times 10^{- 3}$ and z = 1000m. Figure 6(a) shows that the decrease of the value of γ has no significant improvement on the detection probability of the signal OAM mode for WG beams. However, in Fig. 6(b), it is observed that γ has a significant effect on the mode weight of each OAM mode and the mode weight decrease with the increase of γ.

Fig. 6 Detection probability and Mode weight of the signal OAM mode for WG beams against γ for $C_{n}^{2}$ .

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Figures 7(a)-7(b) show the detection probability and the mode weight of the signal OAM mode l₀ = 1 for WG beams versus the real parameter of WG beams W₀ from 0 to $1 \times 10^{- 3}$ for propagation distance z = 1000m, 1500m and 2000m, and parameters λ = 1550nm, w₀ = 0.02m, $C_{n}^{2} = 10^{- 15} m^{3 - α}$ , α = 3.97 and γ = 1. Analogous to γ, the effect of W₀ on the detection probability of the signal OAM mode for WG beams is negligible, but the mode weight decreases with the increase of the value of W₀. These results show that, although γ and W₀ have no effect on the detection probability spectrum, decreasing the values of γ and W₀ is also a good choice to improve the mode weight (energy content) of the signal OAM mode.

Fig. 7 Detection probability and Mode weight of the signal OAM mode for WG beams against W₀ for z.

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4. Conclusions

We have quantitatively described the effects of turbulence on the detection probability spectrum and the mode weight of the OAM for WG beams in non-Kolmogorov turbulence channels. The influence of the OAM quantum number, propagation distance, beamwidth, the refractive-index structure parameter, non-Kolmogorov parameter, wavelength and the real parameters of WG beams γ and W₀ are discussed. Our results show that the detection probability of the signal OAM mode decreases with the increasing of the signal OAM mode quantum number l₀, propagation distance, non-Kolmogorov turbulence parameters and the generalized refractive index structure parameters, and with the decreasing of the wavelength and beamwidth of the beams. The crosstalk probability of OAM mode is symmetrical and has an opposite trend of the detection probability. Although γ and W₀ has no effect on the detection probability of the signal OAM mode, the mode weight decreases with the increasing of the real parameters of WG beams γ and W₀. Compared with LG beam, WG beam is a better light source for mitigating the effects of turbulence on the detection probability of the signal OAM mode in the weak turbulence regime due to its nondiffraction property.

Acknowledgments

This work is supported by the graduate student research innovation project of Jiangsu province general university and Fundamental Research Funds for the Central Universities of China (Grant no. 1142050205135370).

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Influence of atmospheric turbulence on the transmission of orbital angular momentum for Whittaker-Gaussian laser beams

Abstract

1. Introduction

2. The probability distribution of spiral plane modes

3. Numerical discussion

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (7)

Equations (13)

Optics Express