Reducing the cross-talk among different orbital angular momentum modes in turbulent atmosphere by using a focusing mirror

Mengyao Zhou; Yunqin Zhou; Gaofeng Wu; Yangjian Cai

doi:10.1364/OE.27.010280

1. Introduction

The vortex beams carrying OAM keep a subject of intense research because they have important applications in optical manipulation and trapping [1], imaging [2], quantum entanglement [3], free-space optical (FSO) communication [4], fiber optical communication [5] and so on. In particular, significant progress has been made in propagation of vortex beams in turbulent atmosphere. It is well known that the light beam will be disturbed by the random fluctuation of the refraction index of turbulent atmosphere [6–9]. Therefore, there are many OAM states at the received plane when the incident beam with single OAM state propagates in turbulent atmosphere. This is because the initial OAM state will seriously spread to neighbor OAM states. This effect is called as cross-talk of OAM state. How to reduce the cross-talk is still a challenge.

It should be emphasized that there are at least two kinds of manner to be considered to reach the aim. One is the adaptive optics and the other is using special beams. Although the adaptive optics is successfully used to compensate the orbital angular momentum of a vortex beam disturbed by turbulent atmosphere [10,11], its optical system is very complicated. For the special beams, such as the Whittaker-Gaussian beam [12], the autofocusing Airy beam [13], the tailored Airy vortex beam array [14], the Bessel beam and Laguerre–Gaussian beam [15], have been considered to use in FSO communication. They can partially mitigate the effects of turbulence on the detection probability of the signal OAM mode in the weak turbulence regime, but the generation of those beams need to some other complicated instruments. In the present paper, we propose a simple manner but with a good effect to reduce the cross-talk among different OAM modes in turbulent atmosphere.

The paper is organized as follows. Basic derivation for the OAM spectrum of a Gaussian vortex beam in weak turbulence is given in Sec. 2. In Sec. 3, some numerical results and discussion are showed to illustrate the effect of our method. In Sec. 4, the numerical simulation is used to confirm the obtained results. The conclusions of the present paper are given in Sec. 5.

Fig. 1 Schematic of the OAM state disturbed by turbulent atmosphere after passing through a focussing mirror.

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2. Theoretical derivation

The light beam carried OAM propagating in turbulent atmosphere is illustrated in Fig. 1. A light beam with vortex phase passes through a focusing mirror and then propagates in turbulent atmosphere. The source plane is set just behind the mirror. Therefore, the beam profile and phase wavefront in the initial plane (z = 0) can be assumed as [16]

E (r, θ, 0) = exp [- \frac{(1 + i C_{0})}{2 R_{0}^{2}} r^{2} + i l_{0} θ],

where

(r, θ)

is the polar coordinates and l₀ is the quantum number of the OAM mode in the source plane. Here, R₀ is initial beam size,

C_{0} = k R_{0}^{2} / F

is the initial beam prefocusing parameter and F is the focal distance or the distance from the source plane to the plane of the receiver,

k = 2 π / λ

is the wave number with wavelength λ. The constant C₀ occurring in Eq. (1) shows the existence of a mirror. Now, let’s consider this beam propagates in free space. The complex amplitude of this beam in the receiver plane can be obtained with the help of the Huygens-Fresnel integral as

\begin{matrix} E_{f r e e} (ρ, φ, z) = - \frac{i}{λ z} \exp (i k z) \\ \times \int \int E (r, θ, 0) \exp {\frac{i k}{2 z} [r^{2} + ρ^{2} - 2 r ρ \cos (φ - θ)]} r d r d θ, \end{matrix}

where

(ρ, ϕ)

is the polar coordinates. By substituting Eq. (1) into Eq. (2), and utilizing the following expression [17]

\begin{array}{l} \int_{0}^{2 π} \exp [i n φ_{1} - i x \cos (φ_{2} - φ_{1})] d φ_{1} \\ = 2 π (- i)^{n} J_{n} (x) \exp [i n φ_{2}], \end{array}

one can obtain

\begin{matrix} E_{f r e e} (ρ, φ, z) = 2 π {(- i)}^{l_{0}} (- \frac{i}{λ z}) \exp (i k z) \exp (i l_{0} φ) \exp (\frac{i k}{2 z} ρ^{2}) \\ \times \frac{\sqrt{π} k ρ}{8 z α^{3 / 2}} \exp (- \frac{k^{2} ρ^{2}}{8 α z^{2}}) [I_{0.5 l_{0} - 0.5} (\frac{k^{2} ρ^{2}}{8 α z^{2}}) - I_{0.5 l_{0} + 0.5} (\frac{k^{2} ρ^{2}}{8 α z^{2}})], \end{matrix}

where

α = \frac{1}{2 R_{0}^{2}} + \frac{i C_{0}}{2 R_{0}^{2}} - \frac{i k}{2 z}

,

J_{n} ()

and

I_{n} ()

denote the nth order Bessel function and modified Bessel function, respectively.

Then, we consider this beam propagates in turbulent atmosphere. If we restrict ourselves to considering the weak turbulence regime, the cumulative effect of the turbulence over the propagation path can be thought as a pure phase perturbation on the beam at the output plane z. Using the Rytov approximation, the complex amplitude of the Gaussian vortex beam which propagates in weak turbulence at distance z can be represented as [18]

E (ρ, φ, z) = E_{f r e e} (ρ, φ, z) exp [ψ (ρ, φ, z)],

where

ψ (ρ, φ, z)

is the complex phase perturbation of the field due to the atmospheric turbulence along the propagation channel.

In a quantum description, the complex amplitude $E (ρ, φ, z)$ of one light beam can be expressed as the superpositions of states of different OAM [19], i.e.,

E (ρ, φ, z) = \frac{1}{\sqrt{2 π}} \sum_{l = - \infty}^{+ \infty} a_{l} (ρ, z) exp (i l φ),

and the expansion coefficient is

a_{l} (ρ, z) = \frac{1}{\sqrt{2 π}} \int_{0}^{2 π} E (ρ, φ, z) exp (- i l φ) d φ,

where l denotes the quantum number of the OAM mode in the z plane. Based on Eq. (7) and Eq. (5), in paraxial turbulent channels, the probability density

〈 | a_{l} (ρ, z) |^{2} 〉

of OAM mode of one vortex beam is given by the integral

\begin{matrix} 〈 | a_{l} (ρ, z) |^{2} 〉 = \frac{1}{2 π} \int \int 〈 E_{f r e e}^{*} (ρ, φ_{1}, z) E_{f r e e} (ρ, φ_{2}, z) 〉 \\ \times 〈 \exp [ψ^{*} (ρ, φ_{1}, z) + ψ (ρ, φ_{2}, z)] 〉 \exp [- i l (φ_{2} - φ_{1})] d φ_{1} d φ_{2}, \end{matrix}

where the angular brackets represent ensemble average,

〈 \exp [ψ^{*} (ρ, φ_{1}, z) + ψ (ρ, φ_{2}, z)] 〉

is the phase correlation function, and can be given by [20] as

〈 \exp [ψ^{*} (ρ, φ_{1}, z) + ψ (ρ, φ_{2}, z)] 〉 = \exp [- \frac{2 ρ^{2} - 2 ρ^{2} cos (φ_{2} - φ_{1})}{ρ_{0}^{2}}]

In Eq. (9), ρ₀ is the spatial coherence radius, and is given by [20]

ρ_{0} = {[\frac{π^{2}}{3} k^{2} z \int_{0}^{\infty} κ^{3} ϕ_{n} (κ) d κ]}^{- 1 / 2} .

Here, the Tatarskii spectrum for the spatial power spectrum of the refractive-index fluctuation is adopted as [21]

ϕ_{n} (κ) = 0.033 C_{n}^{2} κ^{- 11 / 3} exp (- κ^{2} / κ_{m}^{2}),

where

C_{n}^{2}

is the structure constant of the refractive index fluctuation of the turbulence and

κ_{m} = 5.92 / l_{i}

with l_i being the inner scale of the turbulence. On substituting Eq. (11) into Eq. (10), then we can get

ρ_{0} = {(0.5466 k^{2} z C_{n}^{2} l_{i}^{- 1 / 3})}^{- 1 / 2} .

By substituting Eq. (4) and Eq. (9) into Eq. (8), after complicated integration, we read

\begin{matrix} 〈 | a_{l} (ρ, z) |^{2} 〉 = \frac{π^{4} k^{2}}{8 z^{2} | α^{1.5} |^{2}} {(\frac{1}{λ z})}^{2} ρ^{2} \exp [- \frac{k^{2} ρ^{2}}{8 α^{*} z^{2}} - \frac{k^{2} ρ^{2}}{8 α z^{2}}] \exp [- (\frac{2}{ρ_{0}^{2}}) ρ^{2}] \\ \times I_{l_{0} - l} [(\frac{2}{ρ_{0}^{2}}) ρ^{2}] {| [I_{0.5 l_{0} - 0.5} (\frac{k^{2} ρ^{2}}{8 α z^{2}}) - I_{0.5 l_{0} + 0.5} (\frac{k^{2} ρ^{2}}{8 α z^{2}})] |}^{2} . \end{matrix}

The energy weight for each of the spiral harmonics of the Gaussian vortex beams is defined as the follows

ω_{l} = \int_{0}^{R} 〈 | a_{l} (ρ, z) |^{2} 〉 ρ d ρ,

where R is the radius of the receiving aperture. Then, we can get

\begin{matrix} ω_{l} = \frac{π^{4} k^{2}}{8 λ^{2} z^{4} | α^{1.5} |^{2}} \int_{0}^{R} ρ^{3} \exp [- q ρ^{2}] \exp [- t ρ^{2}] \\ \times I_{l_{0} - l} [t ρ^{2}] {| [I_{0.5 l_{0} - 0.5} (p ρ^{2}) - I_{0.5 l_{0} + 0.5} (p ρ^{2})] |}^{2} d ρ, \end{matrix}

where

t = \frac{2}{ρ_{0}^{2}}

,

p = \frac{k^{2}}{8 α z^{2}}

,

q = \frac{k^{2}}{8 α^{*} z^{2}} + \frac{k^{2}}{8 α z^{2}}

. The normalized energy weight of the OAM mode of the Gaussian vortex beams in the paraxial region can be expressed as

P_{l} = \frac{ω_{l}}{\sum_{l^{'} = - \infty}^{\infty} ω_{l^{'}}},

where

\sum_{l^{'} = - \infty}^{\infty} ω_{l^{'}}

is the total energy. In fact, the OAM spectrum is consisted of different P_l. Based on Eqs. (13), (15) and (16), we can numerically calculate the OAM spectrum.

3. Numerical results and discussion

Here some numerical results are used to illustrate a focusing mirror improving the propagating behavior of OAM. In the examples we set λ = 1060 nm, $R_{0} = 0.02$ m, $C_{n}^{2} = 10^{- 15} m^{- 2 / 3}$ , $l_{i} = 0.001$ m, $z = 1000$ m, $R = 0.05$ m, unless specified otherwise.

Fig. 2 The probability density of OAM mode for different values of C₀ with $F =$ (a) $- 1$ km,(b) infinity and (c) 1 km, respectively.

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Figure 2 shows the variation of the probability density of OAM mode as a function of radial axis for different values of C₀ with $l = 1, 2$ and $l_{0} = 1$ .

Fig. 3 The received OAM spectrum for different quantum number. The first row and second row denote there exists a focusing mirror and does not exist a focusing mirror, respectively.

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It is found that the parameter C₀ has obvious influence on the probability density. In particular, one can detect the OAM state (l = 1) with large probability and small space area when C₀ is taken a positive value (see Fig. 2(c)). This situation forms strong contrast by comparing with that of $C_{0} = 0$ and $C_{0} < 0$ that correspond to the situation of no focusing mirror and of diverging mirror, respectively. At the same time, one can find that the probability difference of detecting the OAM state of l = 1 and l = 2 becomes large when we change C₀ from negative value to zero and to positive value (see the filled area of Fig. 2). It means that the OAM mode crosstalk is reduced by using a focusing mirror. This can be understood as the beam’s interacted area with the turbulence decreases on propagation when the beam is focused. Therefore, adding a focusing mirror can improve the communication quality of the light beam in turbulent atmosphere. To directly see the influence of C₀ on the OAM state in turbulent atmosphere, we will show the OAM spectrum of the Gaussian vortex beam in turbulent atmosphere.

Fig. 4 The received OAM spectrum varying with different valued of R₀ for $C_{0} = 0$ (the first row) and $C_{0} \neq 0$ (the second row).

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Figure 3 plots the normalized energy weight of OAM mode for different quantum number without a focusing mirror $C_{0} = 0$ (the first row) and with a focusing mirror $C_{0} \neq 0$ (the second row), respectively. From the first row, one can find the normalized energy weight of central OAM mode is 0.78, 0.7, 0.63 corresponding to l₀=1, 2, 3, respectively. However, it rises to 0.9, 0.81, 0.74, respectively (see the second row). This value is larger which means the OAM mode is purer, i.e., the cross talk is weaker. Therefore, we can say that the existing of a focusing mirror can reduce the cross-talk among different OAM modes. We should point out that this is the main result of this paper. In addition, one can find that the rate of the normalized energy weight of central OAM mode growth is 15.4%, 15.7% and 17.5% corresponding to l₀=1, 2, 3, respectively. That means the quantum number is larger the action of mirror is stronger.

Fig. 5 The varying of the normalized energy weight of the central OAM mode ( $l = l_{0} = 3$ ) as a function of R₀ for $C_{0} = 0$ and $C_{0} \neq 0$ .

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The influence of the beam size on the OAM spectrum is showed in Fig. 4. One can find that the distribution of the OAM spectrum varies with the changing of beam size (see the second row). For comparing, the OAM spectrum for different values of R₀ without the mirror is plotted in the first row. Comparing the first row with the second row, we find that the mirror can reduce the cross-talk and the action of mirror with a large beam size is strong. For stressing the action of the mirror, the varying of the normalized energy weight of the central OAM mode as a function of R₀ is plotted in Fig. 5. As reported in [20], for the beam without mirror, there exists an optimal value R₀ for the normalized energy weight of the central OAM mode being maximum. However, this situation is different from that of the beam with a mirror that the curve slowly goes up to its asymptotic limit. This is because the increasing of R₀ leads to the increasing of C₀ and the probability density $〈 | a_{l} (ρ, z) |^{2} 〉$ will increase.

Fig. 6 The received intensity and phase (one realization) of the Gaussian vortex beam in turbulent atmosphere for $C_{0} = 0$ and $C_{0} \neq 0$ .

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Fig. 7 The received OAM spectrum (averaged with 200 realizations) of the Gaussian vortex beam in turbulent atmosphere for $C_{0} = 0$ and $C_{0} \neq 0$ .

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4. Numerical simulation

For illustrating the analytical results in above section, we simulate the Gaussian beam with vortex phase propagating in turbulent atmosphere by using the random phase screens [22]. In our simulation, the parameters are set as λ = 1060 nm, $R_{0} = 0.06$ m, $l_{0} = 3$ , $l_{i} = 0.001$ m, $C_{n}^{2} = 10^{- 15} m^{- 2 / 3}$ , F = 1 km. The size of the phase screen is set as 0.22 m with 512×512 pixels. The propagation distance is 1 km and with 13 screens. One can find the performance of turbulence resisting for a vortex beam with mirror is better than that of the beam without the mirror in Fig. 6. For further quantitative comparison of the crosstalk performance between the beam with a mirror ( $C_{0} \neq 0$ ) and without a mirror ( $C_{0} = 0$ ), the received OAM spectrum are depicted in Fig. 7. It is found that the crosstalk of the vortex beam without the mirror ( $C_{0} = 0$ ) is more serious than that of the beam with a mirror ( $C_{0} \neq 0$ ). This result agrees with the conclusion that we discussed in above section.

5. Conclusions

In summary, the propagation of a Gaussian vortex beam with focusing mirror in Kolmogorov turbulence has been derived. The probability density and the normalized energy weight of the received OAM mode were numerically calculated. It was found that the probability of detecting one OAM mode was enhanced and the cross talk of different OAM modes was reduced by adding a focusing mirror. In addition, the source beam size is larger, the action of mirror is stronger. Those results were also confirmed, in principle, by the numerical simulation. Our results will be useful for improving the quality of the FSO communication.

Funding

National Natural Science Fund for Distinguished Young Scholar (11525418); National Natural Science Foundation of China (NSFC) (11604264, 91750201); Natural Science Basic Research Plan in Shaanxi Province of China (Program No. 2017JQ6032); Science Foundation of Northwest University (15NW28).

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Reducing the cross-talk among different orbital angular momentum modes in turbulent atmosphere by using a focusing mirror

Abstract

1. Introduction

2. Theoretical derivation

3. Numerical results and discussion

4. Numerical simulation

5. Conclusions

Funding

References

Cited By

Figures (7)

Equations (16)

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