1 Introduction

Entanglement [1] is one of the most puzzling properties of quantum theory. Entangled photons shared between two distant observers can offer the potential applications in quantum cryptography [2], quantum teleportation [3], and quantum computation [4]. In 2007, Ursin had experimentally demonstrated entanglement-based quantum key distribution over 144 km [5], while the polarization-entangled photon pairs had only two-dimensional Hilbert space. The orbital angular momentum (OAM) states of a photon in principle have an infinite number of Eigen states providing the possibility to store and process quantum information in a higher dimension [6]. Recently, many theoretical and experimental studies have addressed the effect of atmospheric turbulence on the entangled OAM photons [7,8,9,10,11,12,13]. While the entangled OAM photons propagate through the turbulent atmosphere, the OAM entanglement is destroyed by beam wandering, wavefront distortions and scintillation induced by fluctuations of the atmospheric optical refractive index. This is one of biggest challenges for development of quantum communication systems with high information capacity.

As we know, adaptive optics (AO) is an effective technique to mitigate wavefront distortion induced by atmospheric turbulence and improve image quality in classical fields such as laser communication [14, 15], astronomical imaging [16,17,18] and so on.

Recently, it has been experimentally demonstrated that the crosstalk induced by turbulence between OAM channels has been modified under AO compensation condition [19,20,21]. The results of previous research show that the system power penalty can be efficiently improved after adaptive optics compensation when OAM beams propagate through atmospheric turbulence. The influence of the wavelength offset between the Gaussian beacon and the OAM beams on the compensation performance of all multiplexed beams was investigated [22]. In conclusion, previous studies have focused on the effect of atmospheric turbulence on the entangled OAM photons and the compensation of single-channel OAM optical transmission. Entangled OAM photons propagating will involve two different transmission paths, we need to use two beacons for detection and correction for the compensation. The compensation performance needs to be verified, which will make us more interested. In addition, the main factors that affect the performance of AO system, such as wavefront reconstruction algorithm and spatial bandwidth, have not been analyzed in previous studies.

In this study, we explore the performance of the entangled OAM photons propagating though atmospheric turbulence with adaptive optics system compensation under different wavefront reconstruction algorithm and spatial bandwidth of AO system condition. The performance of wavefront reconstruction algorithm (LS and BP) is tested. OAM scattering coefficients is compared between the condition of with and without AO compensation. Furthermore, we show a comparison result of the numerical simulation and theory calculation through theoretical analysis for the concurrence with different spatial bandwidth.

2 Numerical procedure

The schematic drawing of a pair of OAM-entangled qubits propagating through atmospheric turbulence with adaptive optics compensation is shown in Fig. 1. Beacons adopted the ideal Gaussian beam. Beacon beam A and B propagating through atmospheric turbulence arrive to the wavefront sensor, which can measure the wavefront phase difference and wavefront tilt. This phase will be reconstructed, fitted, and applied to the corrector by the controller generating signals. The phase conjugate is applied to the corrector to compensate aberration induced by atmospheric turbulence. A pair of OAM-entangled qubits is generated by the entanglement photon source and reflected by the correctors propagating through turbulence to Detector A and Detector B, respectively.

Fig. 1
figure 1

Schematic diagram of a pair of OAM-entangled qubits propagating through atmospheric turbulence with adaptive optics compensation system

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2.1 Adaptive optics compensation model

We modeled the main components of an AO system including wavefront sensor, reconstructor, and corrector. The wavefront sensor is modeled by an ideal phase difference sensor [23]. In the numerical simulation, a two-dimensional array of phase differences between neighboring nodes of the computational grid in the directions of the x- and y-axes. The elements of phase differences array \(\Delta\) can be, respectively, written as

$$\begin{gathered} \Delta {x_{i,j}}=\arg ({u_{i+1,j}}u_{{i,j}}^{*}) \hfill \\ \Delta {y_{i,j}}=\arg ({u_{i,j+1}}u_{{i,j}}^{*}), \hfill \\ \end{gathered}$$
(1)

where arg() is an operator that takes the phase of a complex-valued quantity and \(u\) is complex amplitude.

In a standard adaptive optics system, wavefront reconstruction generally utilizes a least mean square error wavefront reconstructor (LS). The least-squares estimate of phase is given by [24]

$${\phi _{{\text{LS}}}}={({\Gamma ^T}\Gamma )^{ - 1}}\Gamma \Delta ,$$
(2)

where \(\Gamma\) is geometry matrix depending on the phase determination from wavefront slopes.

Since LS algorithm will not be able to properly determine the phase of the optical field in the vicinity of a branch point; many methods have been looked at to modify this problem, such as the Goldstein algorithm [25], Fried’s exponential algorithm [26] and Bigot’s branch-point reconstruction algorithm [27] (BP) etc. BP algorithm is adopted in the study. The core of BP algorithm is to find the eigenvector corresponding to maximum eigenvalue of reconstruction matrix \(II\), which can be expressed as

$${(IIu)_j}=\sum\limits_{k} {{w_{j,k}}} \exp (i{\Delta _{j,k}}){u_k}.$$
(3)

The phase \(\phi\) is obtained through

$$\phi =\arg (u).$$
(4)

The phase \(\phi\) still cannot to be directly applied to phase corrections onto a corrector as there are wrapping cuts and branch cuts existing in the phase \(\phi\). It is necessary to unwrap the phase. We use the Todd M. Venema’s method [28]. The phase \(\phi\) is divided into two parts: LS components of field \({\phi _{{\text{LS}}}}\) and non-LS components of the field \({\phi _{{\text{NonLS}}}}\). The components \({\phi _{{\text{LS}}}}\) can be obtained by Eq. (2). \({\phi _{{\text{NonLS}}}}=\phi - {\phi _{{\text{LS}}}}\) is restricted to some \(2\pi\) range to let the normalized cut length be minimized. The finally unwrapped phase can be expressed as

$${\phi _{{\text{unwrapped}}}}={\phi _{{\text{LS}}}}+{\phi _{{\text{NonLS(min(cut))}}}}.$$
(5)

The most widely used corrector is a continuous face sheet deformable mirror (DM). In simulation, an ideal deformable mirror is modeled by directly calculating the conjugated phase of Eq. (5). To further analyze the impact of spatial bandwidth on compensation performance, three different elements DM are simulated including 19-element, 61-element and 127-element DM.

The phase \(S\) induced by the deform mirror can be expressed as

$$S(x,y)=\sum\limits_{{i=1}}^{N} {{V_i}I{n_i}(x,y)} ,$$
(6)

where \(N\) is the actuator number of the DM, \({V_i}\) represents the \(i\)th actuator control voltage and \(I{n_i}(x,y)\) is the influence function of DM. The control voltage of the DM actuator can be solved by LS algorithm

$${V_i}={(I{n^T}In)^{ - 1}}In\phi ,$$
(7)

where \(\phi\) is the phase reconstructed by the wavefront sensor. The influence function of DM \(I{n_i}(x,y)\) can be described as

$$I{n_i}(x,y)=\exp \left[ {\ln {p_{\text{d}}}\frac{{{{(x - {x_i})}^2}+{{(y - {y_i})}^2}}}{{r_{{\text{d}}}^{2}}}} \right],$$
(8)

where \(({x_i},{y_i})\) denotes the location coordinates of the \(i\)th actuator, \({p_{\text{d}}}\) represents the coupling factor between the adjacent actuators, and \({r_{\text{d}}}\) is the distance between the adjacent actuators. The diagrams of the actuator location coordinate for 19-element, 61-element and 127-element deformable mirror are shown in Fig. 2.

Fig. 2
figure 2

The diagrams of the actuator location coordinate for 19-element, 61-element and 127-element deformable mirror

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2.2 Numerical model of entangled states propagation

We assumed that input state is Bell state encoded by LG modes with the opposite azimuthal quantum number, which is expressed as

$${\left| \psi \right\rangle _{{\text{in}}}}=\frac{1}{{\sqrt 2 }}\left( {{{\left| \ell \right\rangle }_{\text{A}}}{{\left| { - \ell } \right\rangle }_{\text{B}}}+{{\left| { - \ell } \right\rangle }_{\text{A}}}{{\left| \ell \right\rangle }_{\text{B}}}} \right),$$
(9)

where the subscripts A and B are labeled as the two different paths, and \(\ell\) is the azimuthal mode order. The input state is divided into four optical fields. \({\left| \ell \right\rangle _{\text{A}}}\) and \({\left| { - \ell } \right\rangle _{\text{A}}}\) share one turbulence path A; \({\left| l \right\rangle _{\text{B}}}\) and \({\left| { - l} \right\rangle _{\text{B}}}\) share another turbulence path. For the input state we only consider the zero radial index \(p=0\) and azimuthal mode of \(\ell =1,3,5\).

Then the propagation of the optical field is simulated by using split step FFT algorithm. In the simulation, the atmospheric turbulence along the path is also modeled by turbulent phase screens, which power spectrum functions obeying Kolmogorov spectrum statistics [8, 30]. The phase function for each phase screen can be expressed as

$$\phi \left( {x,y} \right)={k_0}\int_{0}^{{\Delta z}} {\delta n\left( {x,y,z} \right)} {\text{d}}z,$$
(10)

where \({k_0}\) is the wave number in vacuum. The power spectral density of refractive index fluctuations in simulation is given by

$${\Phi _n}(\kappa )=0.033C_{n}^{2}{({\kappa ^2}+\kappa _{0}^{2})^{ - 11/6}}{\text{exp(}} - {\kappa ^2}{\text{/}}\kappa _{m}^{2}{\text{)}}.$$
(11)

where \(C_{n}^{2}\) (in units of \({m^{ - 2/3}}\)) is refractive-index structure parameter that describes the strength of the turbulence along the path, \(\kappa\) is the magnitude of three dimensional wave number vector, \({\kappa _0}={{2\pi } \mathord{\left/ {\vphantom {{2\pi } {{L_0}}}} \right. \kern-0pt} {{L_0}}}\), \({\kappa _m}={{2\pi } \mathord{\left/ {\vphantom {{2\pi } {{l_0}}}} \right. \kern-0pt} {{l_0}}}\), \({L_0}\) and \({l_0}\) are the outer scale and inner scale of turbulence, respectively. The turbulent phase screens can be realized by spectral method in which phase screens are randomly generated by the fast Fourier transform (FFT) associated with power density spectrum [33].

After each free-space propagation step, the output quantum state \(\left| {{\psi _{m^{\prime\prime}}}} \right\rangle\) is obtained from the beam profiles. The ensemble average of the density matrices corresponding to \(M\) different instances of the medium, which is given by [8, 9]

$$\rho =\frac{{\sum\nolimits_{{m^{\prime\prime}}}^{M} {\left| {{\psi _{m^{\prime\prime}}}} \right\rangle \left\langle {{\psi _{m^{\prime\prime}}}} \right|} }}{{Tr\{ \sum\nolimits_{{m^{\prime\prime}}}^{M} {\left| {{\psi _{m^{\prime\prime}}}} \right\rangle \left\langle {{\psi _{m^{\prime\prime}}}} \right|} \} }}.$$
(12)

The concurrence is used to describe the decay of entanglement, for a maximally entangled state, C = 1, and for a non-entangled state, C = 0. The concurrence of a mixed bi-photon states is quantified in terms of Wootter’s formula (12)

$$C\left( \rho \right)=\hbox{max} \left\{ {0,\sqrt {{\lambda _1}} - \sqrt {{\lambda _2}} - \sqrt {{\lambda _3}} - \sqrt {{\lambda _4}} } \right\},$$
(13)

where \(\lambda\) are non-negative eigen values, in decreasing order of the matrix

$$\xi =\rho \left( {{\sigma _{\text{y}}} \otimes {\sigma _{\text{y}}}} \right){\rho ^ * }\left( {{\sigma _{\text{y}}} \otimes {\sigma _{\text{y}}}} \right),$$
(14)

with the asterisk being the complex conjugate and \({\sigma _{\text{y}}}\) being the Pauli \(y\) matrix.

3 Results and discussion

3.1 Parameters’ description and setting

The effect of the turbulence on the entanglement of a photon pair is completely described by a single dimensionless parameter given by \({\omega _0}/{r_0}\). ω0 is the waist width of the optical beam and r0 is the Fried parameter corresponding to the spatial coherence of the distortions, which is given by

$${r_0}={(0.423{k^2}C_{n}^{2}z)^{ - 3/5}},$$
(15)

where \(k=2\pi /\lambda\) is the optical wave number, \(C_{n}^{2}\) is the structure constant of the turbulence, \(z\) is the propagation distance. The variance of the log-amplitude fluctuations can be used as a measure of the strength of the atmosphere scintillation, which for a plane wave propagating through turbulence is expressed as [11]

$$\sigma _{\chi }^{2}=0.307C_{n}^{2}{k^{7/6}}{z^{11/6}},$$
(16)

In the simulation, the key physical parameters are listed in Table 1.

Table 1 Physical parameters of the simulation
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3.2 Simulation results

We first investigate the performance of the AO compensation with ideal corrector when single-photon state propagating in the turbulence scene. Assume the input state is \(\left| \ell \right\rangle =\left| { - 1} \right\rangle\). The intensity and phase distribution of output state’s wave function without compensation and with LS and BP compensation are shown in Fig. 3. It is easy to find that the stronger the turbulence is, the more distorted the intensity and phase of beams will be. The intensity and phase distribution are well-stayed with BP algorithm.

Fig. 3
figure 3

Intensity and phase distribution of output state’s wave function after propagating through turbulence without compensation and with LS and BP compensation. a With turbulence \({{{\omega _0}} \mathord{\left/ {\vphantom {{{\omega _0}} {{r_0}}}} \right. \kern-0pt} {{r_0}}}=0.4\). b With turbulence \({{{\omega _0}} \mathord{\left/ {\vphantom {{{\omega _0}} {{r_0}}}} \right. \kern-0pt} {{r_0}}}=2.8\). c With turbulence \({{{\omega _0}} \mathord{\left/ {\vphantom {{{\omega _0}} {{r_0}}}} \right. \kern-0pt} {{r_0}}}=4.0\)

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We measured the mode scattering of output state as a function of \({{{\omega _0}} \mathord{\left/ {\vphantom {{{\omega _0}} {{r_0}}}} \right. \kern-0pt} {{r_0}}}\) with initial state \({\left| \ell \right\rangle _A}={\left| { - 1} \right\rangle _A}\) shown as diagrams in Fig. 4. The result is obtained through ensemble average of 500 turbulence steps. With the increase of \({{{\omega _0}} \mathord{\left/ {\vphantom {{{\omega _0}} {{r_0}}}} \right. \kern-0pt} {{r_0}}}\), the output state is generically scattered away from the input state, first to the neighboring state and then to state further away. The output state of scattering is more quick without compensation. This scattering process is slowed down with AO compensation. BP compensation has better performance. Combined with Figs. 3 and 4, we find the output state of scattering is more closely related to phase distribution than intensity.

Fig. 4
figure 4

Mode scattering under different turbulence conditions with initial state \(\left| \ell \right\rangle =\left| { - 1} \right\rangle\)

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The decay of entangled bell state (Eq. 9) as a function of \({{{\omega _0}} \mathord{\left/ {\vphantom {{{\omega _0}} {{r_0}}}} \right. \kern-0pt} {{r_0}}}\) for \(\ell =1,3,5\) is calculated and the result is shown in Fig. 5. The ideal corrector is used. The density matrices Eq. (12) of output states are obtained from ensemble averages of the 500 propagation steps. The concurrence expressed in Eq. (13) is computed using Eq. (12). One can see that the concurrence of output states is improved under adaptive optics compensation conditions for all \(\ell =1,3,5\). Compared to BP algorithm, the concurrence always decreases quickly when \({{{\omega _0}} \mathord{\left/ {\vphantom {{{\omega _0}} {{r_0}}}} \right. \kern-0pt} {{r_0}}}\) > 2.50 with LS algorithm.

Fig. 5
figure 5

The concurrence of output states versus turbulence strength

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Figure 6 shows the Rytov number [31] (log-amplitude variance of plane wave) \(\sigma _{\chi }^{2}\) as a function \({{{\omega _0}} \mathord{\left/ {\vphantom {{{\omega _0}} {{\rho _0}}}} \right. \kern-0pt} {{\rho _0}}}\), under same turbulence condition as in Fig. 5. We can see BP algorithm has better performance than LS algorithm when \(\sigma _{\chi }^{2}\) > 0.2. This is due to the branch-points generated when Rytov number is close to 0.2. This result is consistent with other results in the literature [31, 32]. Combined with Figs. 5 and 6, the BP algorithm has better performance even under strong scintillation conditions.

Fig. 6
figure 6

Rytov number verse \({{{\omega _0}} \mathord{\left/ {\vphantom {{{\omega _0}} {{r_0}}}} \right. \kern-0pt} {{r_0}}}\)

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\(\Theta \left( {r,\Delta l} \right)\) is the scattering coefficients between azimuthal modes for optical power in an annulus of radius \(r\) [34],

$$\Theta \left( {r,\Delta l} \right)=\frac{1}{{2\pi }}\int_{0}^{{2\pi }} {{C_\phi }(r,\Delta \theta )\exp \left[ { - i\Delta l\Delta \theta } \right]d\Delta \theta } ,$$
(17)

where the scattering coefficients depend on the difference \(\Delta l=l - {l_0}\), which represents the resultant OAM spreading from the initial eigenvalue \({l_0}\). \({C_\phi }(r,\Delta \theta )\) is referred to as the rotational correlation function of the phase perturbations at radius \(r\), which is given by

$${C_\phi }\left( {r,\Delta \theta } \right)=\exp \left[ { - \frac{1}{2}{D_\phi }\left( {\left| {2r\sin \left( {\frac{{\Delta \theta }}{2}} \right)} \right|} \right)} \right].$$
(18)

For Kolmogorov turbulence-phase statistics, \({D_\phi }(r)=6.88{\left( {r/{r_0}} \right)^{5/3}}\)the rotational coherence function at radius \(r\) is

$${C_\phi }\left( {r,\Delta \theta } \right)=\exp \left[ { - 6.88 \times {2^{2/3}}{{\left( {\frac{r}{{{r_0}}}} \right)}^{5/3}}{{\left| {\sin \left( {\frac{{\Delta \theta }}{2}} \right)} \right|}^{5/3}}} \right].$$
(19)

We suppose that the spatial bandwidth of AO system is mainly limited by DM. The DM can be used as a high-pass spatial filter with the spatial frequency threshold\(\Lambda =2\pi /2{r_d}\). The phase structure function \({D_{\phi \_AO}}(r)\) with AO compensation can be described as

$$\begin{aligned} D_{{\phi \_Ao}} \left( r \right) & = 2.91k^{2} \int_{0}^{L} {C_{n}^{2} (z)\int_{0}^{\infty } {\kappa ^{{ - 8/3}} (1 - J_{0} (\kappa r))U(\kappa - \Lambda ){\text{d}}\kappa {\text{d}}z} } \\ & = 2.91k^{2} \int_{0}^{L} {C_{n}^{2} (z)dz} \\ & \quad \times \left( {1.118r^{{5/3}} - \frac{3}{{5\Lambda ^{{(5/3)}} }}\left( {_{1} F_{2} ( - 5/6;1/6,1;1/4r^{2} \Lambda ^{2} } \right) - 1} \right), \\ & = 6.88\left( {\left( {\frac{r}{{r_{0} }}} \right)^{{5/3}} - c_{1} \left( {\frac{{r_{d} }}{{r_{0} }}} \right)^{{5/3}} \left( {1 - _{1} F_{2} ( - 5/6;1/6,1;\frac{{\pi ^{2} r^{2} }}{{4r_{d}^{2} }}} \right)} \right). \\ \end{aligned}$$
(20)

where \(U(x)\) is unit step function, \(_{1}{F_2}\) is generally a hypogeum function and \({c_1}\) is a constant \({c_1}=\frac{{12\Gamma (11/6){2^{2/3}}}}{{5{\pi ^{5/3}}\Gamma ( - 5/6)}}\).

The rotational correlation function with AO compensation can be obtained by substituting Eq. (20) into Eq. (28), which is given by

$$\begin{aligned} {C_{\phi \_Ao}}\left( {r,\Delta \theta } \right) & =\exp \left[ { - \frac{1}{2}\left( {6.88 \times {2^{5/3}}{{\left( {\frac{r}{{{r_0}}}} \right)}^{5/3}}{{\left| {\sin \left( {\frac{{\Delta \theta }}{2}} \right)} \right|}^{5/3}}} \right.} \right. \\ & \quad - \left. {\left. {{c_1}{{\left( {\frac{{{r_d}}}{{{r_0}}}} \right)}^{5/3}}\left( {1{ - _1}{F_2}( - 5/6;1/6,1;\frac{{{\pi ^2}{r^2}{{\left| {\sin \left( {\frac{{\Delta \theta }}{2}} \right)} \right|}^{5/3}}}}{{r_{d}^{2}}}} \right)} \right)} \right]. \\ \end{aligned}$$
(21)

According to the theory above, the results of comparing the first few OAM scattering coefficients \(\Theta \left( {r,\Delta l} \right)\) to \(r/{r_0}\) in the condition of without-compensation (Fig. 7a) and with AO compensation (Fig. 7b–c) are shown Fig. 7. For \(r<<{r_0}\), the effects of the phase aberrations are weak and the OAM scattering is small, but they increase rapidly as \(r\) becomes comparable to \({r_0}\). With AO compensation, the OAM scattering is effectively weakened, and the smaller the drive spacing, the more obvious the scattering weakening is.

Fig. 7
figure 7

OAM scattering coefficients \(\Theta \left( {r,\Delta l} \right)\) between models \({l_0}\) and \({l_0} \pm \Delta l\) for an annulus of radius in a beam for aberrations with Fried parameter \({r_0}=0.05\,\,{\text{m}}\) in the condition of a without compensation, b with AO compensation \({r_d}=0.1\,\,{\text{m}}\), c with AO compensation \({r_d}=0.05\,\,{\text{m}}\) and d with AO compensation \({r_d}=0.01\,\,{\text{m}}\)

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To further study the decoherence of entangled states propagating through turbulence atmosphere with different AO compensation spatial bandwidth, the concurrence expression of output states with AO compensation is derived based on the expression without compensation. The concurrence of output states without compensation can be expressed as [7]

$$C=\hbox{max} \left( {0,\frac{{1 - b/a}}{{{{(1+b/a)}^2}}}} \right),$$
(22)

where

$$\begin{aligned} a & ={\left| {\left\langle {{{\psi _{ \pm \ell }}}} \mathrel{\left | {\vphantom {{{\psi _{ \pm \ell }}} { \pm \ell }}} \right. \kern-0pt} {{ \pm \ell }} \right\rangle } \right|^2} \\ & =\frac{1}{{2\pi }}{\int_{0}^{\infty } {r\left| {{R_{l,0}}(r)} \right|} ^2}{\text{d}}r\int_{0}^{{2\pi }} {\text{d}} \theta \exp \left[ - 0.5{D_\phi }\left( {2r\left| {\sin \frac{\theta }{2}} \right|} \right)\right], \\ \end{aligned}$$
(23)
$$\begin{aligned} b & = \left| {\left\langle {\psi _{{ \pm \ell }} |\, \mp \ell \,} \right\rangle } \right|^{2} \\ & = \frac{1}{{2\pi }}\int_{0}^{\infty } {r\left| {R_{{l,0}} (r)} \right|^{2} } {\text{d}}r\int_{0}^{{2\pi }} {\text{d}} \theta e^{{ - i\theta \Delta l}} \exp \left[ { - 0.5D_{\phi } \left( {2r\left| {\sin \frac{\theta }{2}} \right|} \right)} \right], \\ \end{aligned}$$
(24)

where \({R_{l,0}}(r)\) is the radial part of the input LG beam.

Figure 8 represents a comparison between the results of numerical simulation and theory calculation which is obtained by substituting Eq. (20) into Eq. (22). The BP algorithm be adapted in numerical simulation. From these figures, we find concurrence of output states decay more slowly using 127-element deformable mirror than 61-element. High spatial bandwidth has better compensation effect. Meanwhile, the results of numerical simulation are basically consistent with theoretical results.

Fig. 8
figure 8

The concurrence of output states versus turbulence strength

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4 Conclusion

In this paper, an adaptive optics compensation system for the entangled OAM photon propagating in turbulence is described, in which two Gaussian beams are used to detect the wavefront distortions induced by the atmospheric turbulence for compensating the OAM beams. The model of entangled OAM propagating in turbulence with adaptive optics compensation is established. The influence of reconstruction algorithm and spatial bandwidth of the corrector is analyzed. The results show that BP algorithm has better compensation performance when Rytov number exceeds 0.2 than LS algorithm. When using the same reconstruction algorithm, the correction effect is largely determined by the spatial bandwidth of the corrector. The analytical relation expression of the scattering coefficients and the concurrence with the spatial bandwidth under adaptive optics compensation conditions are derived, which can characterize the correction limits of adaptive optical systems with different spatial bandwidth. The study can provide reference for AO compensation of entangled OAM communication in atmospheric turbulence. However, the AO system model is relatively ideal and simpler. The influence that noises have on wavefront sensor and bandwidth of control system need further study. And the non-Kolmogorov turbulence should be further considered.