Turbulence heterodyne coherent mitigation of orbital angular momentum multiplexing in a free space optical link by auxiliary light

Chunyong Yang; Chuang Xu; Wenjun Ni; Yu Gan; Jin Hou; Shaoping Chen

doi:10.1364/OE.25.025612

1. Introduction

It is well known that orbital angular momentum (OAM) multiplexing has been identified as a bright prospect for increasing the optical transmission capacity and spectrum efficiency [1–6]. In free space optical (FSO) communication, a level of 100 terabit free-space data transmission based on OAM multiplexing has been demonstrated [7]. Although FSO communication remains great potential, the atmospheric turbulence is a major challenge that may lead to the serious degradation in the bit error rate (BER) performance of the system and even make the communication link infeasible [8]. Light beams carrying orbital angular momentum (OAM) possess a helical azimuthal phase pattern of $\exp (i l θ)$ , in which $θ$ represents the azimuth angle and $l$ determines the OAM value ( $l$ is also defined as the topological charge of the OAM states) [9]. In principle, unlimited values of $l$ denote infinite OAM states, and each OAM state is mutually orthogonal. As a result of the existence of turbulence, the helical phase fronts of OAM beams are distorted by atmospheric turbulence resulting in low received power and crosstalk between the neighboring OAM modes [10–15].

In order to mitigate the influence of atmospheric turbulence, various mitigation techniques have been proposed, such as aperture averaging technique, diversity technique, equalization techniques, channel coding, adaptive optics, etc [8,16]. Recently, channel coding and adaptive optics are commonly used to mitigate turbulence effects in OAM-multiplexed FSO communication link. For example, Z. Qu and I.B. Djordjevic proposed a turbulence mitigation scheme by a low-density parity check (LDPC)-coded orbital angular momentum (OAM)-based transmission system [17,18]. G. Xie et al. proposed a phase correction method using a Zernike-polynomials-based stochastic-parallel-gradient-descent algorithm. They use the intensity pattern of OAM beam to derive the phase correction pattern [19]. S. Li et al. experimentally demonstrate turbulence compensation for free-space four-fold and eight-fold 16-ary quadrature amplitude modulation (16-QAM) carrying orbital angular momentum (OAM) multicasting links by using an adaptive feedback correction technique [20]. Meanwhile, S. Zhao et al. proposed and analyzed both channel coding and adaptive optics schemes, and they acquired that the BER performance have greatly improved by using both channel code and wavefront correction method [21,22]. Though the method of channel coding can improve the BER performance in a certain way, it has a limitation in turbulence mitigation and is hard to deal with the strong turbulence situation. The scheme of adaptive optics is commonly adopted to mitigate the turbulence effects, because it improves the performance of the FSO communication link to a certain extent. Nevertheless, it is still imperfect since the adaptive optics is applied to realize the wavefront correction which contains a wavefront sensor, wavefront corrector and wavefront controller [23–25]. Therefore, an adaptive optics system has a highly complex structure which is difficult for the practical applications.

In this paper, a novel turbulence mitigation scheme using a special coherent detection is proposed. Compared with the traditional coherent detection, the special coherent detection is characterized by using the common-path auxiliary light as the local oscillation light. The scheme does not need to employ some adaptive optics components or any other independent correction modules. Furthermore, after matching the corresponding spatial phase pattern of the common-path auxiliary light, the information-carrying OAM beam is able to be demodulated. As a consequence, the counteraction of turbulence distortions and the de-multiplexing of OAM beams can be achieved simultaneously based on the scheme stated above. In numerical simulations, the quadrature phase shift keying (QPSK) modulation is adopted. Additionally, the OAM channels of multiplexing transmission are selected as $l = 2$ , $l = - 3$ , $l = 5$ and $l = - 8$ . The bit error rate (BER) is able to achieve 10⁻⁵ even under strong turbulence conditions (the atmospheric refractive index structure function $C_{n}^{2} > 1.0 \times 10^{- 13} m^{- 2 / 3}$ ). Finally, the influence of beam size to the proposed method is analyzed and two adjustment methods are also discussed.

2. Principles of the proposed turbulence mitigation method

Figure 1 illustrates the principles of turbulence mitigation for the OAM-based FSO link. The transmitter loads the QPSK signal over the OAM beams which are combined together with the Gaussian beam by a polarization beam splitter 1 (PBS 1). Accompanied by the auxiliary Gaussian light, the multiplexed OAM beams are then transmitted through the same emulated atmospheric turbulence (AT). The light is divided into two beams by a polarization beam splitter 2 (PBS 2), and then the two different polarization states beams are formed at the receiver. After turbulence, one beam used as the signal light carries OAM modes. The Gaussian beam is used as the local oscillation light after appending a special phase pattern by means of a spatial light modulator (SLM N, N = 1,2,..., N<20) [26,27]. Then the two beams interfere at the beam splitter N (BS N). The output response current is obtained by the photo detector N (PD N, N = 1,2,..., N<20). Each SLM N is imprinted with a corresponding helical phase pattern $\exp (i l_{N} θ)$ . At the same time, each phase pattern is corresponding to a specific topological charge value. In this way, N OAM modes can be successfully de-multiplexed. To verify it, both the theoretical analysis and numerical simulations are conducted.

Fig. 1 Schematic diagram of the proposed turbulence mitigation in OAMs multiplexing link. BS, beam splitter; PBS, polarization beam splitter; SLM, spatial light modulator; AT, atmospheric turbulence; PD, photo detector.

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In the theoretical analysis, an OAM beam $U (r, θ)$ can be formed by attaching a spiral phase mask $\exp (i l θ)$ to a Gaussian beam. It is described as

U (r, θ) = A (r) \cdot \exp (i l θ)

where

A (r) \propto \exp (- r^{2} / w_{0}^{2})

is the complex electric field amplitude at the waist of the Gaussian beam,

r

is the radial distance starting from the center axis of light beam, and

w_{0}

is the diameter of the waist. After modulating QPSK signal over N OAM beams, the hybrid light field of N multiplexed OAM beams is given by [4]

U_{M U X} (r, θ, t) = \sum_{s = 1}^{N} m_{s} (t) \cdot A_{s} (r) \cdot \exp (i l_{s} θ)

where

l_{s}

is the topological charge of OAM beam which is distincted by the corner mark of the

s

,

A_{s} (r)

represents the amplitude of the complex electric field and is various with different

l_{s}

.

m_{s} (t)

represents the QPSK signal of OAM channel s, which can be written as

m_{s} (t) = α \cdot \exp (ω \cdot t + i ψ_{s}),_{} ψ_{s} \in {\frac{π}{4}, \frac{3 π}{4}, \frac{5 π}{4}, \frac{7 π}{4}}

where

α

is the amplitude of signal and

ω

is the angular frequency of OAM beams.

ψ_{s}

denotes the carried phase information of OAM channel s, and t is the time axis.

To simulate real-time atmospheric turbulence, a Kolmogorov turbulence phase screen model [14,28–30] is taken into account. Its power spectrum function can be expressed as

Φ (κ) = 0.033 C_{n}^{2} κ^{- 113}, κ \in [2 π / L_{0}, 2 π / l_{0}]

where

C_{n}^{2}

is the atmospheric refractive index structure function.

L_{0}

represents the outer turbulence radius, and

l_{0}

is the inner turbulence radius. In addition, the three important parameters of the Rytov variance

σ_{R}^{2}

, the overall strength of turbulence

D / r_{0}

and the Fresnel number

N_{f} = D / (λ L)

are selected to emulate the atmospheric turbulence [13]. Here,

λ

, L,

D

respectively represents the wavelength, link distance and the aperture sizes of the link. According to the reference [31], the Fried coherence length

r_{0}

can be revealed by the following equation:

r_{0} = {[\frac{2.91}{6.88} k^{2} \int_{0}^{L} C_{n}^{2} (z) d z]}^{- 3 / 5}

Where

z

represents the actual propagation distance of the light beam. Assuming

C_{n}^{2} (z)

is a constant value along the path, that is,

C_{n}^{2} (z) \equiv C_{n}^{2}

. k is the propagation constant which is equal to

2 π / λ

. Thus, the expression of

r_{0}

can be simplified as

r_{0} = 0.185 {[λ^{2} / (C_{n}^{2} \cdot L)]}^{3 / 5}

. To measure the scintillation, the Rytov variance

σ_{R}^{2}

is given by the following expression [15]

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

In this way, a Kolmogorov turbulence phase screen can be expressed as

D_{ϕ} (r) = 6.88 {(r / r_{0})}^{5 / 3}

_$ϕ$ is assumed to the distortion phase caused by turbulence after the light beam transmits through the turbulence phase screen. Thus, the received light field of OAM beams and the Gaussian beam can be respectively written as

U_{M U X}^{'} (r, θ, t) = \sum_{s = 1}^{N} m_{s} (t) \cdot A_{s}^{'} (r) \cdot \exp (i l_{s} θ) \cdot \exp (i ϕ)

and

U (r, t) = A^{'} (r) \cdot \exp (i ϕ)

_{$A_{s}^{'} (r)$} and

A^{'} (r)

respectively represent the distorted intensity of OAM beams and Gaussian beam. Then appending the specific spiral phase pattern

\exp (i l_{p} θ)

to the

U (r, t)

, then Gaussian beam can be expressed as

U^{'} = A^{'} (r) \cdot \exp (i ϕ) \cdot \exp (i l_{p} θ)

In coherent detection, the photocurrent $I$ is proportional to the following equation:

I \propto \int_{0}^{+ \infty} \int_{0}^{2 π} ({| U_{M U X}^{'} + U^{'}^{*} |}^{2}) r d r d θ

where * represents the symbol of conjugate transformation. If we take the Eqs. (8) and (10) into Eq. (11), and then filter out the direct current components, the result can be derived as follows:

\begin{matrix} I = β Re (\int_{0}^{+ \infty} \int_{0}^{2 π} U_{M U X}^{'} \cdot U^{' *} r d r d θ) \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \\ = β Re {\begin{cases} \int_{0}^{+ \infty} \int_{0}^{2 π} [\sum_{s =1}^{N} m_{s} (t) \cdot A_{s}^{'} (r) \cdot \exp (i l_{s} θ) \cdot \exp (i ϕ)] \cdot A^{'}^{*} (r) \\ \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} \times \exp (- i ϕ) \cdot \exp (- i l_{p} θ) r d r d θ \end{cases}} \end{matrix}

where

β

is the constant,

Re (\cdot)

represents the function of real-part extraction. It is easy to see that the distorted phase pattern

\exp (i ϕ)

can be counteracted in expression (12). The photocurrent expression can be simplified as the following equation when the corner mark s is equal to the p.

I = {\begin{cases} β \cdot Re [_{\int_{0}^{+ \infty} \int_{0}^{2 π} m_{p} (t) \cdot A_{p}^{'} (r) \cdot A^{'^{*}} (r) \cdot r d r d θ_{}]} {_{}}_{} i f_{}_{} s = p; \\ \begin{matrix} \begin{matrix} \begin{matrix} 0 & \begin{matrix} \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} \end{matrix} o t h e r w i s e \end{matrix} \end{matrix} \end{matrix} \end{cases}

If the amplitude coefficient of photocurrent is set as

c_{p}

, and the OAM channel is corresponding to p, the photocurrent can be given as (14) after extracting the real-part.

I = c_{p} \cdot \cos (Δ ω \cdot t + ψ_{p})

Here,

Δ ω

represents the frequency difference between the signal light and the local oscillation light. In the heterodyne coherent detection, a band pass filter is used to filter out the frequencies beyond the range of

Δ ω

. Thus,

ψ_{p}

can be successfully demodulated without the distorted phase

ϕ

. Additionally and importantly, it can be derived from the Eqs. (12) and (13) that the amplitude coefficient

c_{p}

is distinguishable to each other. Consequently, the multiplexed OAM modes are promising to be separated.

3. Numerical simulations and discussion

In numerical simulations, Laguerre-Gaussian (LG) mode is taken as a research object of OAM mode. The electric field of a LG mode is given by [32]

\begin{array}{l} U (r, θ, z) = \sqrt{\frac{2 p!}{π (p + | l |)!}} \frac{1}{w (z)} {[\frac{r \sqrt{2}}{w (z)}]}^{| l |} L_{p}^{l} [\frac{2 r^{2}}{w^{2} (z)}] \\ \begin{matrix} \begin{matrix}  \end{matrix} & \times \exp [\frac{- r^{2}}{w^{2} (z)}] \end{matrix} \exp [\frac{- i k r^{2} z}{2 (z^{2} + z_{R}^{2})}] \\ \begin{matrix} \begin{matrix}  \end{matrix} & \times \exp [i (2 p + | l | + 1) \tan^{- 1} \frac{z}{z_{R}}] \end{matrix} \exp (- i l θ) \end{array}

where

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

is the beam radius with propagation distance of

z

.

z_{R} = π w_{0}^{2} / λ

is the Rayleigh length. Here, the generalized Laguerre polynomial

L_{p}^{l} (\cdot)

is equal to 1 if the radial mode number

p = 0

. All the OAM modes are selected as LG mode to implement our method. It is noteworthy that the LG mode can be written as

{LG}_{p l}

(here, subscript

p

and

l

are the parameters aforementioned).

To deal with the turbulence effects, two common phase retrieval algorithms are mentioned in references [33,34], which requires to measure or estimate the distorted phase. Hence, a wavefront sensor or corrector is essential to phase retrieval in their method. Unlike these two phase retrieval algorithms, the proposed method is based on the principle of two-beam distortion counteraction in Fig. 2. $U_{{LG}_{02}}$ and $U_{G}$ are the initial light field of LG₀₂ beam and Gaussian beam, respectively. Similarly, $U_{{LG}_{02}}^{'}$ and $U_{G}^{'}$ stands for the after-turbulence light field, respectively. In coherent detection, the distorted phase $ϕ (x, y)$ is counteracted by the integral calculation $〈 U_{{LG}_{02}}^{'}, U_{G}^{'^{*}} 〉$ . Note that, $〈 a, b 〉$ means doing the integral calculation $\int_{0}^{+ \infty} \int_{0}^{2 π} ({| a + b |}^{2}) r d r d θ$ in polar coordinate $(r, θ)$ , where a, b are arbitrary variables. Take the LG₀₂ mode as an example, the LG₀₂ beam and Gaussian beam are multiplexed in two orthogonal polarization directions.The phase distortion caused by turbulence is regarded as a whole part along the path. The phase variation of the two beams is identical. What’s more, the distorted phase $ϕ$ of the two beams is counteracted in the coherent detection instead of using wavefront sensor or corrector. It should be noticed that the SLM in Fig. 2 acts as a phase matching function. That is, the detected power of LG₀₂ mode can be enhanced by imprinting the helical phase pattern $e x p (i \cdot 2 θ)$ on the SLM. Thus, the other detected modes are restrained because of the orthogonality. To validate the proposed method, four types of simulation links are conducted in the following work, that is, single mode link, double-modes link, three-modes link and four-modes link. In all simulations, a 1 km transmission link with aperture of the link $D = w (z) \approx 5_{} cm$ is selected. Besides, the wavelength of $λ = 1550_{} nm$ is corresponding to a Fresnel number of $N_{f} = 32$ . The parameter $C_{n}^{2}$ is set as $1 \times 10^{- 15} ~ 1 \times 10^{- 13}_{} m^{- 2 / 3}$ to simulate weak to strong atmospheric turbulence. For easier comparison, all of the power values have been normalized and the power weight is defined as $P_{l} = d_{l} / \sum_{l = - 10}^{10} d_{l}$ ( $d_{l}$ is the obtained power values of each OAM mode). In other words, the power weight is equivalent to the proportion of each OAM mode in total power. To clearly describe the results, the power weight comparison at the condition of initial, after-turbulence and detected for LG₀₂ mode are given in Table 1. $P_{initial}$ , $P_{after-tur .}$ and $P_{detected}$ represent initial, after-turbulence and detected power weight, respectively.

Fig. 2 Schematic diagram of two-beam distortion counteraction in LG₀₂ mode transmission.

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Table 1. Power weight of LG₀₂ mode in four simulations

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Firstly, a single LG₀₂ mode transmission link simulation is illustrated in Fig. 3. To exemplify better performance of the proposed method, strong turbulence condition ( $C_{n}^{2} = 1 \times 10^{- 13}_{} m^{- 2 / 3}$ ) is selected in this single mode transmission link. And the Rytov variance $σ_{R}^{2}$ is 1.99 by calculating the Eq. (5) and $D / r_{0}$ equals to 2.532. Figure 3(a) shows the initial normalized power spectrum of LG₀₂ mode before transmission. After turbulence, the normalized power spectrum of the LG₀₂ mode is shown in Fig. 3(b), which indicates the serious distortion leading to an indistinguishable power spectrum. By using the proposed method, the LG₀₂ mode can be clearly distinguished with 50% enhancement in normalized power displayed in Fig. 3(c) and Table 1. In addition, the normalized power gain is also pronounced under moderate turbulence situation ( $C_{n}^{2} = 1 \times 10^{- 14}_{} m^{- 2 / 3}$ ). Combined with the theoretical analysis, the performance can be dramatically improved owing to the counteraction of the distorted phase pattern $\exp (i ϕ)$ .

Fig. 3 (a) Normalized power spectrum of the initial OAM, where the single LG₀₂ mode is transmitted through the strong turbulence ( $C_{n}^{2} = 1 \times 10^{- 13}_{} m^{- 2 / 3}$ ) in a distance of 1 km. (b) Normalized power spectrum of LG modes after turbulence. (c) Normalized power spectrum of the detected LG modes.

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Secondly, double OAM modes (LG₀₂ and LG₀₄) in a multiplexing communication link are presented in Fig. 4. Under the multiplexed condition, a QPSK signal is modulated on both modes LG₀₂ and LG₀₄. Besides, considering the crosstalk between each mode, a moderate turbulence situation with $C_{n}^{2} = 1 \times 10^{- 14}_{} m^{- 2 / 3}$ is chosen for a better performance. According to the calculation results, the parameter of $σ_{R}^{2}$ equals to 0.199 and $D / r_{0}$ is 0.636. Similar to the single-mode link, LG₀₂ beam and LG₀₄ beam are multiplexed in X polarization direction, while Gaussian beam is in Y polarization direction. The most obvious difference between single-mode link and double-mode link lies in the phase matching method. In double-modes link, the phase matching method is realized by two SLM imprinted with two specific computer-generated hologram for $l = 2$ and $l = 4$ . In this way, the OAM modes can be successfully demodulated and the QPSK signal is subsequently obtained. As a result, the normalized power spectrum of initial OAM modes, OAMs after turbulence and detected OAM modes are respectively given by Figs. 4(a)-4(c). In the figures, the power weight of LG₀₂ mode and LG₀₄ mode are eventually dominant from the chaotic spectrum induced by turbulence. As for other detected modes, their spectrums appear large restriction. It is verified that our phase matching method is effectively. In addition, about $50 %$ enhancement in normalized power of the detected object modes is demonstrated, which is comparable to simulation 1. The inset of the constellation diagrams in Fig. 4(c) shows that the BER can reach to $1 \times 10^{- 5}$ with the endurance of even stronger turbulence conditions.

Fig. 4 (a) Normalized power spectrum of the initial state of two OAM multiplexed modes (LG₀₂ and LG₀₄) is transmitted through the moderate turbulence ( $C_{n}^{2} = 1 \times 10^{- 14}_{} m^{- 2 / 3}$ ) for a distance of 1 km. (b) Normalized power spectrum of LG modes after turbulence. (c) Normalized power spectrum of the detected LG modes.

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Thirdly, three-modes link (LG₀₂, LG_0-3 and LG₀₄) is also demonstrated. The principle of this three-modes link is shown in Fig. 1 when N equals to 3. Also the detailed schematic is similar to the double-modes link. Observed from the results in Fig. 5, the proposed method keeps working well in three-modes link under the moderate turbulence ( $C_{n}^{2} = 1 \times 10^{- 14}_{} m^{- 2 / 3}$ ). In Figs. 5(b) and 5(c), though the power enhancement of the detected LG₀₂, LG_0-3 and LG₀₄ are uneven, the distinction of the detected power weight among the objective LG modes and others is unambiguous. The inset of the constellation diagrams in Fig. 5(c) reveals that the BER can reach to $1 \times 10^{- 5}$ in spite of more OAM modes transmission.

Fig. 5 (a) Normalized power spectrum of the initial state of three OAM multiplexed modes (LG₀₂, LG_0-3 and LG₀₄) is transmitted through the moderate turbulence ( $C_{n}^{2} = 1 \times 10^{- 14}_{} m^{- 2 / 3}$ ) for a distance of 1 km. (b) Normalized power spectrum of LG modes after turbulence. (c) Normalized power spectrum of the detected LG modes.

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Finally, the system BER is analyzed when the QPSK signal is imprinted over four OAM channels (LG₀₂, LG_0-3, LG₀₅ and LG_0-8) under the same condition of the two-mode transmission link. In QPSK modulation, $1 \times 10^{6}$ sampling points are evenly distributed in phase of $π / 4$ , $3 π / 4$ , $5 π / 4$ and $7 π / 4$ . Via the moderate turbulence, the received constellation diagram distributions and BER curves of four OAM channels are displayed in Fig. 6(a). It shows that the BER decreases with the topological charge increasing. The reason is that the larger of the absolute value of topological charge is, the more serious of the distortion will be. The system performance of all the OAM channels is much better than the limited BER of the forward error coding ( $3.8 \times 10^{- 3}$ ). To further analyze the BER performance, the atmospheric turbulence strength $C_{n}^{2}$ is selected as an variable ranging from $1 \times 10^{- 15}_{} m^{- 2 / 3}$ to $1 \times 10^{- 13}_{} m^{- 2 / 3}$ . Figure 6(b) demonstrates that the stronger of the atmospheric turbulence is, the worse of the BER of four OAM channels will be. It can be seen from the tendency of BER curves that the proposed method is fit for weak-to-moderate turbulence mitigation. Moreover, the result indicates that the lower-order channel has the better performance.

Fig. 6 BER of four multiplexed OAM channels carrying the QPSK signal. (a) The BER and the constellation diagram of different OAM channels. Here, the absolute value $| l | = 2, 3, 5, 8$ correspond to $l = 2, - 3, 5, - 8$ , respectively. (b) The BER of four multiplexed OAM channels under different atmospheric turbulence strength.

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This phenomenon may attribute to the different beam size between the OAM modes and Gaussian auxiliary mode. That is to say, the lower-order OAM beams experience smaller turbulence distortion since their beam sizes increase slower than those of the higher-order OAM beams [25]. The non-overlap between OAM modes and Gaussian auxiliary mode may affect the phase distortion counteraction. And the negative influence on phase distortion counteraction would become more serious in higher-order OAM modes if beams size differs a lot.

To clearly see the superiority of the proposed method, a simulation comparison between the proposed method and the direct detection has been conducted. In the direct detection, the multiplexed OAM beams are transmitted through turbulence while the auxiliary Gaussian beam is not. Thus, the received photocurrent of the direct detection component is given by

I_{d i r e c t} = \int_{0}^{+ \infty} \int_{0}^{2 π} {{| U_{M U X}^{'} +[U_{G} \cdot \exp (i l_{p} θ)]^{*} |}^{2}} r d r d θ .

Here, U_G represents the light field of initial Gaussian beam without turbulence-induced distortion, and

U_{M U X}^{'}

denotes the multiplexed OAM beams with turbulence distortion. It can be seen from Fig. 7, the proposed method shows strongly resistance for the turbulence strength of

C_{n}^{2} = 1 \times 10^{- 14}_{} m^{- 2 / 3}

, whereas the direct detection without distorted auxiliary light seems more serious. The direct detection scheme has an ambiguous constellation diagram even under the weak turbulence condition. The reason is that the initial Gaussian beam does not have the distorted phase caused by atmospheric turbulence. Thus, the phase distortion cannot be eliminated by the direct detection. Therefore, the proposed method is valid to deal with the phase distortion.

Fig. 7 The constellation diagram of four multiplexed OAM channels carrying the QPSK signal. Both the proposed method and the direct detection are presented.

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In our further study, it has been demonstrated that the turbulence mitigation is affected with the overlap size between OAM modes and the Gaussian auxiliary mode. But it should not obliterate the positive effect of our proposed method. It is prone to think directly of overcoming the adverse effect of the overlap size by beam size adjustment. Moreover, there are two ways to adjust the beam adaptation. The one is to adjust Gaussian mode beam size to fit for OAM beams, the schematic diagram of which is presented in Fig. 8(b), the other is to adjust OAM beams waist sizes to fit for Gaussian mode beam, the schematic diagram of which is presented in Fig. 8(c). For ease of comparison, Fig. 8(a), which does not make any adjustments, is also ploted. The specific effect of both adjustments is discussed as follows.

Fig. 8 Schematic diagram of three situations. (a) without adjustment; (b) adjust the Gaussian beam size to fit for the highest order OAM beam; (c) adjust the OAM beam waists to fit for the Gaussian beam. w_OAM, r_OAM are respectively the beam waists, beam sizes of the OAM beams, and w_G, r_G are respectively the beam waist, beam size of the Gaussian beam.

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In the scheme of Gaussian mode beam size fitting for OAM beams in Fig. 8(b), the initial waists of all beams are defined as w₀. The Gaussian beam is expanded to fit for the maximum beam size of OAMs. Based on the references [35–37], the relationship between the topological charge $l$ and the beam radius $r (z)$ is defined as $r (z) = \sqrt{(| l | + 1) / 2} \cdot w (z)$ . Here, $r (z)$ represents the square root of the radial variance of the intensity distribution, and $w (z) = w_{0} \sqrt{1 + {(z / z_{R})}^{2}}$ is the beam radius with propagation distance of $z$ . If w₀ is set to 0.05m and $z_{R} = π w_{0}^{2} / λ$ is the Rayleigh length, note that, $w (z)$ is approximately equal to $w_{0}$ while the propagation distance is 1 km, and then the beam size of LG₀₂, LG_0-3, LG₀₅ and LG_0-8 can be respectively obtained as 0.061m, 0.071m, 0.0866m, and 0.106m. Thus, we set the beam size of Gaussian mode as 0.106 m to fit for LG_0-8. Comparisons of the constellation diagram between with and without Gaussian beam size adjustment scheme are shown in Figs. 9(a) and 9(b). To clearly observe the distinctions, four-modes (LG₀₂, LG_0-3, LG₀₅ and LG_0-8) beams are transmitted under the same turbulence condition $C_{n}^{2} = 5 \times 10^{- 14}_{} m^{- 2 / 3}$ . As is shown in Fig. 9(b), the BER of higher-order OAM beams (LG₀₅ and LG_0-8) are obviously improved with Gaussian beam size adjustment.

Fig. 9 The constellation diagram of four multiplexed OAM channels based on the proposed method (a) without adjustment, (b) adjust the Gaussian beam size to fit for the highest order OAM beam and (c) adjust the OAM beam waists to fit for the Gaussian beam.

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In the scheme of OAM beams adjustment for fitting Gaussian beam in Fig. 8 (c), all the OAM beams have equal size and their beam waists are correspondingly adjusted. Here, referring to the maximum beam size stated above, all the OAM beam sizes are maintained as $r (z) = r_{G} = {0.106}_{} m$ . Then we get the expression of $w (z) = \sqrt{2 / (| l | + 1)} \cdot r (z) = r_{G} \cdot \sqrt{2 / (| l | + 1)}$ . After adjustment for fiting Gauss beam, the waist of LG₀₂, LG_0-3, LG₀₅ and LG_0-8 are respectively of 0.087 m, 0.075 m, 0.061 m, 0.05 m. Comparisons of the constellation diagram between with and without OAM beam waists adjustment are presented in Figs. 9(a) and 9(c). To clearly observe the distinctions, four-modes (LG₀₂, LG_0-3, LG₀₅ and LG_0-8) beams are also transmitted under the same turbulence condition $C_{n}^{2} = 5 \times 10^{- 14}_{} m^{- 2 / 3}$ . As it should be seen from Fig. 9(c), the BER of higher-order OAM beams (LG₀₅ and LG_0-8) are obviously improved with OAM beam waists adjustment.

Furthermore, the BER performance could be obtained in Fig. 10. The BER of four OAM channels are all improved by the adjustment schemes, and both of the schemes are effective to solve the difference of the beam sizes. Moreover, by comparing the inserted constellation diagrams in the case of $l = 5$ , the OAM beams adjustment outperforms the Gaussian beam adjustment. But the former is more difficult to be implemented due to its multiple beam variations.

Fig. 10 The BER of four multiplexed OAM channels under the turbulence strength $C_{n}^{2} = 5 \times 10^{- 14}_{} m^{- 2 / 3}$ . The situations of without adjustment, with Gaussian beam size adjustment, and with OAM beam waists adjustment are presented.

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4. Conclusion

In summary, a novel scheme is proposed to transmit the OAM beams together with the Gaussian beam. The OAM beams and Gaussian beam are respectively used as the signal light and local oscillation light in the coherent detection. Besides, a phase matching method is put forward to demodulate each OAM channels. In this way, the turbulence effects are mitigated. Thus, the communication performance is improved. Furthermore, the system complexity is also simplified. Both the single OAM link and the multiple OAM links have been demonstrated in numerical simulation. The result shows that the BER of OAMs transmission can reach to $1 \times 10^{- 5}$ . The normalized power spectrum of the detected OAMs also shows a significant improvement. Additionally, by adjusting the beam size between the OAM modes and auxiliary Gaussian mode, it seems an effective way to further improve the system performance. Our work may pave a way to the practical application in phase distortion compensation and the proposed method is promising to reduce OAM mode crosstalk.

Funding

Natural Science Foundation of China (NSFC) (61002013, 11504435); Natural Science Foundation of Hubei Province (2014CFA051); Key Technology R/D Program of Hubei Province (2015BCE048).

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Project Name	$C_{n}^{2}$ m^-2/3	$P_{initial}$	$P_{after-tur .}$	$P_{detected}$
Single mode Simulation (LG₀₂)	1 × 10⁻¹³	1	0.1449	0.3090
Single mode Simulation (LG₀₂)	1 × 10⁻¹⁴	1	0.2497	0.4758
Double-modes Simulation (LG₀₂ and LG₀₄)	1 × 10⁻¹⁴	0.5	0.1477	0.3479
Three-modes Simulation (LG₀₂, LG_0-3 and LG₀₄)	1 × 10⁻¹⁴	0.333	0.1203	0.3015
Four-modes Simulation (LG₀₂, LG_0-3, LG₀₅ and LG_0-8)	1 × 10⁻¹⁴	0.25	0.0832	0.2835

Turbulence heterodyne coherent mitigation of orbital angular momentum multiplexing in a free space optical link by auxiliary light

Abstract

1. Introduction

2. Principles of the proposed turbulence mitigation method

3. Numerical simulations and discussion

4. Conclusion

Funding

References and links

Cited By

Figures (10)

Tables (1)

Equations (16)

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