# Mitigating the effect of atmospheric turbulence on orbital angular momentum-
based quantum key distribution using real-time adaptive optics with phase
unwrapping

Zhiwei Tao  1,2,3 Yichong Ren  2,4 Azezigul Abdukirim  2 Shiwei Liu  1,2 and
Ruizhong Rao2  1School of Environmental Science and Optoelectronic Technology,
University of Science and Technology of China, Hefei 230026, China
2Key Laboratory of Atmospheric Optics, Anhui Institute of Optics and Fine
Mechanics, Chinese Academy of Sciences, Hefei 230031, China
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract

Quantum key distribution (QKD) employed orbital angular momentum (OAM) for
high-dimensional encoding enhances the system security and information
capacity between two communication parties. However, such advantages
significantly degrade because of the fragility of OAM states in atmospheric
turbulence. Unlike previous researches, we first investigate the performance
degradation of OAM-based QKD by infinitely long phase screen (ILPS), which
offers a feasible way to study how adaptive optics (AO) dynamically corrects
the turbulence-induced aberrations in real time. Secondly, considering the
failure of AO while encountering phase cuts, we evaluate the quality
enhancement of OAM-based QKD under moderate turbulence strengths by AO after
implementing the wrapped cuts elimination. Finally, we simulate that, with
more realistic considerations, real-time AO can still mitigate the impact of
atmospheric turbulence on OAM-based QKD even in the large wind velocity
regime.

††journal: oe††articletype: Research Article

##  1 Introduction

Quantum key distribution (QKD)[1, 2, 3, 4, 5, 6] provides unconditionally
secure random numbers[7, 8, 9, 10, 11, 12] between two authenticated distant
parties (Alice and Bob) based on the no-cloning principle of quantum
physics[13, 14]. In conventional QKD schemes, photons carrying spin angular
momentum are usually used to encode information[15, 16, 17, 18, 19, 20].
However, the information capacity of this binary system is limited to $1$ bit
per photon, making it more vulnerable to eavesdroppers’ attacks. Unlike spin
angular momentum, the orbital angular momentum (OAM)[21] of light, associated
with $l$ intertwined helical phase $e^{il\phi}$ of each photon in a beam,
where $l$ takes integer values and is unbounded, offers an alternative way to
encode information in an infinitely large Hilbert space and improve the
information capacity of QKD system to more than $1$ bit per photon[22].
Moreover, a higher-order system possesses a higher security threshold[23, 24,
25], such as, for the BB84 protocol in dimension $5$, this threshold increases
to the limit of $21\%$ compared to $11\%$ for dimension $2$.

Over the past decade, most previous experiments, including indoors[26, 27, 28,
29, 30, 31, 32, 33, 34, 35] and outdoors[36, 37, 38, 39, 40, 41, 42], have
demonstrated such encoding schemes are feasible up to dimension $7$[30] and
the range as far as $340m$[38]. Nevertheless, the presence of atmospheric
turbulence remains the greatest challenge to effectively implement an OAM-
based QKD experiment in free space. Random air refractive index fluctuations
caused by atmospheric turbulence disrupt the phase alignment of original
transmitted optical fields, split the optical vortex into several individual
vortices and create the photonic OAM pairs during the propagation[43, 44, 45],
all combinations of which result in mode scrambling at the receiver[46, 47,
48, 49, 50, 51, 52, 53], making the high dimensional encoding lose its unique
advantages.

To mitigate the adverse effects of atmospheric turbulence, the common
compensation strategy is to use adaptive optics (AO) [38, 54, 55, 56, 57, 58].
Recently, many numerical experiments have been performed using AO correction
in quantum levels[59, 60, 61]. However, all these scenarios are simulated by
random phase screens, which is only valid if turbulence satisfies Taylor’s
hypothesis[62] (i.e., when the laser pulse width is significantly narrower
than the time scale of random refractive index fluctuations, atmospheric
turbulence is assumed to be stationary during this tiny time fraction). To
overcome this barrier, we employ the infinitely long phase screen (ILPS)[63]
method to dynamically simulate the evolution of the average quantum bit error
rate and the secret key rate of OAM-based QKD.

With the implementation of atmospheric propagation, the branch points[64]
caused by intensity modulation will occur in the phase distribution at the
receiver, broadening the OAM spectrum distribution[45, 65] and leading to a
reduced performance of OAM-based QKD. These phase discontinuities complicate
the phase correction in an AO system. Considering the inability of traditional
AO while encountering phase cuts[66, 67, 68], we evaluate the performance
enhancement of OAM-based QKD by AO after implementing the wrapped cuts
elimination. We also demonstrate that such localized manipulation is
physically possible and brings a significant improvement compared to the
previously poor correction.

It should be noted that this approach brings the drawback that it reduces the
response rate of AO system[54, 69] and tends to affect the correction
performance of them while turbulence is changing rapidly[70]. Concretely, the
slope of beacon light measured by a Hartmann wavefront sensor[71] is not
converted into the voltage signal of a deformable mirror directly but is used
to reconstruct the phase of beacon light. Then, the reconstructed phase is
repaired by phase unwrapping algorithm and finally converted into the voltage
signal of a deformable mirror. To examine the feasibility of our scheme, we
re-evaluate the performance enhancement achieved by AO and conclude that this
scheme is still able to mitigate the impact of atmospheric turbulence on OAM-
based QKD, even in the large wind (we use the word "wind" in general referring
to the crosswind) velocity regime.

This paper is structured as follows. Sec. 2 briefly introduces the two
mutually unbiased bases (MUBs) used in OAM-based QKD and presents the
principle of OAM-based QKD that integrates the effects of atmospheric
turbulence. The main methods used in this paper, including modified ILPS,
perturbed phase correction, and wrapped cuts elimination, are detailed in Sec.
3. In Sec. 4, we first investigate the undesirable impact of atmospheric
turbulence on a single OAM state; secondly, we evaluate the correction effect
of our enhanced AO to compensate the turbulence-induced aberrations for OAM-
based QKD under arbitrary turbulence strengths and correction orders; finally,
we consider the impact of realistic noise contributions on the performance of
OAM-based QKD. At last, Sec. 5 discusses some deficiencies of our numerical
methods and concludes this paper.

##  2 Physical model

###  2.1 Photonic source

A photon’s degrees of freedom, such as time-energy, OAM, and position
momentum, are typically physical sources for implementing high-dimensional
encoding. For an OAM-based protocol, we study the two MUBs that consist of a
group of successive OAM photon states and its complementary Fourier conjugate
angular basis (i.e., so-called angular position (ANG) state)
$\left|j\right\rangle=1/\sqrt{d}\sum_{l=-L}^{L}\left|l\right\rangle\exp\left(-i2\pi
jl/\left(2L+1\right)\right)$, where $\left|l\right\rangle$ represents a
single-photon state of a Laguerre-Gaussian (LG) mode[72]
$LG_{0,l}\left(r,\phi,0\right)$, $d=2L+1$ is the dimension of the encoding
subspace and $L$ is the maximum azimuthal index used in this protocol. Without
loss of generality, the LG modes at output plane $z$ can be described in
normalized cylindrical coordinates by

$\displaystyle LG_{p,l}\left(r,\phi,z\right)$ $\displaystyle=$
$\displaystyle\frac{A}{w_{z}}\left(\frac{\sqrt{2}r}{w_{z}}\right)^{\left|l\right|}L_{p}^{\left|l\right|}\left(\frac{2r^{2}}{w_{z}^{2}}\right)$
(1)
$\displaystyle\times\exp\left[-\frac{r^{2}}{w_{z}^{2}}+i\left(\frac{kr^{2}}{2R_{z}}+l\phi-\left(2p+\left|l\right|+1\right)\varphi_{g}\right)\right],$

with $A=\sqrt{2p!/\pi\left(p+\left|l\right|\right)!}$ representing the
normalization constant and radial quantum number $p$ (For more quantum
properties of $p$, we refer the reader to Refs. [78, 79, 80, 81]), where
$L_{p}^{\left|l\right|}\left(\cdot\right)$ is the generalized Laguerre
polynomial, $w_{z}=w_{0}\sqrt{1+\left(z/z_{R}\right)^{2}}$ is the beam waist
with $w_{0}$ being the beam waist at input plane. $z_{R}=\pi
w_{0}^{2}/\lambda$ and $k=2\pi/\lambda$ denote the Rayleigh range and the wave
number respectively, $\lambda$ is the wavelength,
$R_{z}=z\left[1+\left(z_{R}/z\right)^{2}\right]$ is the radius of curvature
and $\varphi_{g}$ stands for the Gouy phase associated with propagation phase
in this protocol.

Figure 1: (a) The photonic source in Alice’s laboratory randomly produces a
quantum state in either of two mutually unbiased bases and sends it to Bob
through a turbulent atmosphere (modeled by multiple layers of ILPS arranged as
ladder-shaped distribution) toward a detector. The phase-screen array is
ladder-shaped distributed because the divergence angle of LG modes becomes
larger with the increase of propagation distance while holding $\omega_{0}$ of
the LG modes constant[83]. (b) A straightforward illustration for the
principle of ILPS, where the three matrices from left to right illustrate how
the last-generated phase screen connects with the next one. The new column $X$
(illustrated by the notation $\boxdot$) calculated from the partial data $Z$
of last-generated phase screen (illustrated by the notation $\odot$) is added
into the last-generated one (see the second matrix). To prevent the oversized
phase screen, one has to discard the excess part that is useless (i.e., the
column outside the dashed line of the second matrix).

###  2.2 Principle of OAM-based QKD

In the weak scintillation regime, the distortion of a photonic OAM state
$\left|l_{0}\right\rangle$ propagated through atmospheric turbulence can be
considered as a pure phase perturbation[46, 47]. To investigate the influence
of turbulence on OAM-based QKD under arbitrary scintillation conditions, the
split-step method[82] is used to simulate the procedure of atmospheric
propagation instead of the pure phase perturbation approximation. As shown in
Fig. 1(a), multiple layers of ILPS are arranged consecutively to form a phase-
screen array for simulating the propagation of two MUBs across the turbulence.
For convenience of presentation, the unitary operator
$\widehat{U}_{turb}^{\left(i\right)}$ is used to represent the $i$-th
realization of a photonic state traveling through turbulence instead of the
extended Huygens-Fresnel integral[60, 61, 84]. Hence, after undergoing the
unitary transformation, the received state becomes a superposition of several
LG modes, which can be described using ket-bra notation as follows[60, 61]

$\left|\psi_{l_{0}}^{\left(i\right)}\right\rangle=\widehat{U}_{turb}^{\left(i\right)}\left|l_{0}\right\rangle=\sum_{p=0}^{\infty}\sum_{l=-\infty}^{\infty}c_{pl}^{\left(i\right)}\left|pl\right\rangle,$
(2)

with $\left|pl\right\rangle$ corresponding to a single photon state of a LG
mode $LG_{p,l}\left(r,\phi,z\right)$, where
$c_{l,p}^{\left(i\right)}=\left\langle
pl\right|\widehat{U}_{turb}^{\left(i\right)}\left|l_{0}\right\rangle$.
Notably, since LG modes with same $w_{0}$ form an orthonormal basis, the waist
of LG modes are modified to $w_{z}$[85] during the spectral decomposition for
attributing the intermodal crosstalk to the impact of turbulence entirely.
Besides, the probability of finding one photon in the received state with an
azimuthal index $l$ can be written as[86]

$p_{l}^{\left(i\right)}=\sum_{p=0}^{\infty}\left|\left\langle
pl\right|\widehat{U}_{turb}^{\left(i\right)}\left|l_{0}\right\rangle\right|^{2},$
(3)

Based on the principle of a single OAM state propagation, we transfer our
attention to a concrete QKD scheme. Since Alice randomly chooses her photonic
OAM states from two MUBs, we only concern the spectral broadening of initially
transmitted modes within the subspace of primary encoding basis and regard the
mode scrambling outside the subspace as the atmospheric losses during the
simulation. To get the measured matrix of OAM-based QKD, we normalize the
crosstalk probability of each incident OAM state, which can be expressed in
the coordinate representation as[73]

$p_{l_{k}\rightarrow
l_{s}}^{\left(i\right)}=\frac{\sum_{p=0}^{\infty}\int_{0}^{R}\int_{0}^{2\pi}\left|\psi_{l_{k}}^{\left(i\right)}\left(r,\phi,z\right)LG_{p,l_{s}}^{\ast}\left(r,\phi,z\right)\right|^{2}rdrd\phi}{\sum_{l_{s}=-L}^{L}\sum_{p=0}^{\infty}\int_{0}^{R}\int_{0}^{2\pi}\left|\psi_{l_{k}}^{\left(i\right)}\left(r,\phi,z\right)LG_{p,l_{s}}^{\ast}\left(r,\phi,z\right)\right|^{2}rdrd\phi},$
(4)

where $p_{l_{k}\rightarrow l_{s}}^{\left(i\right)}$ represents the probability
of finding OAM $l_{s}$ component from the results of projective measurement
when the incident OAM value equals to $l_{k}$, and $R$ is the radius of
receiving aperture.

In order to explore the impact of atmospheric turbulence on the performance of
OAM-based QKD, this simulation ignores the noises arising from the detector
and experimental instruments, such as dark counts[29], afterpulsing
effect[87], to name just a few. Hence, the bit error arises from the turbulent
channel can be quantified by the average quantum bits error rate (QBER). For
the given two MUBs, the average QBER in OAM basis is[88]

$Q_{OAM}^{\left(i\right)}=\frac{1}{d}\sum_{\begin{subarray}{c}l_{k},l_{s}=-L\\\
k\neq s\end{subarray}}^{L}\sum_{p=0}^{\infty}\left|\left\langle
pl_{s}\right|\widehat{U}_{turb}^{\left(i\right)}\left|l_{k}\right\rangle\right|^{2},$
(5)

Likewise, the average QBER in ANG basis $Q_{ANG}^{\left(i\right)}$ can also be
acquired from the same procedure. By averaging the QBER over two MUBs, the
total errors caused by atmospheric turbulence are described as

$Q^{\left(i\right)}=\frac{1}{2}\left(Q_{OAM}^{\left(i\right)}+Q_{ANG}^{\left(i\right)}\right),$
(6)

Finally, we evaluate the information capacity that is securely exchanged
between both parties before the postselection for OAM-based QKD, quantified by
the minimum secret key rate, which can be calculated as[25, 88]

$r_{\min}^{\left(i\right)}=\log_{2}d+2\left[Q^{\left(i\right)}\log_{2}\frac{Q^{\left(i\right)}}{d-1}+\left(1-Q^{\left(i\right)}\right)\log_{2}\left(1-Q^{\left(i\right)}\right)\right],$
(7)

##  3 Numerical methods

###  3.1 Modified infinitely long phase screen

The simplest method to simulate atmospheric turbulence is to use the random
phase screen[89], together with a low-spatial frequency compensation[90].
However, the random phase screen is established on the fact that laser pulse
width is significantly narrower than the time scale of random refractive index
fluctuations[62]. Therefore, the dynamical process of incident light
propagating through atmospheric turbulence and how AO implements dynamical
correction to the perturbation cannot be investigated within this method
(e.g., the execution rate of AO system is greater than the variation time of
turbulence). Moreover, the static assumption will lose its usefulness when
studying the influence of wind velocity on an AO-aided OAM-based QKD.

The ILPS method gives a feasible solution to overcome this barrier. Without
loss of generality, we summarize and divide the procedure of ILPS’s execution
into two steps as follows[63]: Firstly, we generate an initially random phase
screen and calculate the pixel number (i.e., rows or columns) to be moved (we
call this as the need-to-be-moved pixel number) in each iteration according to
the wind velocity $v$, wind direction $\theta$ and iteration time (i.e., time
interval between two iterations). Secondly, for the $i$-th realization of
turbulence and $j$-th need-to-be-moved pixel, we calculate the new need-to-be-
added row or column $X$ from the partial data $Z$ of last-generated phase
screen, add it into the last-generated one and discard the excess part of the
oversized phase screen (a more straightforward illustration is presented in
Fig. 1(b)). The relationship between $X$ and $Z$ is given by[63]

$X=AZ+B\beta,$ (8)

where $Z$ represents the last $N_{col}$ rows/columns of last-generated phase
screen, $\beta$ is a Gaussian random vector with zero mean and its covariance
equal to unity. The matrix $A$ and $B$ (see more details for the derivation of
$A$ and $B$ and their dimension informations in Ref. [63]) can be calculated
from the covariance of the $X$ and $Z$ vectors, including the matrices
$\left\langle ZZ^{T}\right\rangle$, $\left\langle XZ^{T}\right\rangle$,
$\left\langle ZX^{T}\right\rangle$ and $\left\langle XX^{T}\right\rangle$.
Concretly, all of covariance matrices can be acquired from constructing the
distance matrix $r_{m,n}\equiv R\left(X_{m},Z_{n}\right)$ and acting them on
the phase covariance function $C_{\varphi}\left(r\right)$, where $X_{m}$,
$Z_{n}$ and $R\left(\cdot\right)$ denote the $m$-th element in $X$, the $n$-th
element in $Z$ and the distance function respectively. If the turbulence
spectrum follows the von-Karman rules, $C_{\varphi}\left(r\right)$ can be
expressed as[91]

$C_{\varphi}\left(r\right)=\left(\frac{2\pi
r}{L_{0}}\right)^{\frac{5}{6}}\left(\frac{L_{0}}{r_{0}}\right)^{\frac{5}{3}}\frac{\Gamma\left(11/6\right)}{2^{5/6}\pi^{8/3}}\left[\frac{24}{5}\Gamma\left(\frac{6}{5}\right)\right]^{\frac{5}{6}}K_{5/6}\left(\frac{2\pi
r}{L_{0}}\right),$ (9)

with the outer scale $L_{0}$ and the Fried parameter $r_{0}$, where
$K_{5/6}\left(\cdot\right)$ is the McDonald function and
$\Gamma\left(\cdot\right)$ is the gamma function.

It is worth noting that the conventional ILPS (we use the word "conventional"
in general referring to the ILPS mentioned in Ref. [63]) can only simulate the
effects of wind velocity changes in both horizontal and vertical directions,
which significantly limits the utilizing of this algorithm. For this reason,
we modify the conventional ILPS by decomposing the wind velocity into
horizontal and vertical directions. For simplicity, we now detail our
modifications into two parts: On the one hand, our algorithm first detects
whether a new column needs to be added in the horizontal direction, and if so,
subtracts one pixel number from the horizontally need-to-be-moved pixel number
after each motion (if we assume the need-to-be-moved pixel number equals to
$N_{move}$, then the horizontally and vertically need-to-be-moved pixel number
equal to $N_{move}\sin\theta$ and $N_{move}\cos\theta$ respectively), then,
executes the same procedure in the vertical direction and repeats the two
steps until no motion demand is required in either direction. On the other
hand, after undergoing above manipulation, the residual pixel number of two
directions (i.e., the difference between the horizontally/vertically need-to-
be-moved pixel number and the physically moved pixel number in two
directions), which is commonly less than one pixel number, is added to the
horizontally and vertically need-to-be-moved pixel number for the next
iteration[92] (see two examples of modified ILPS in Fig. 2).

Figure 2: Two independent examples of modified ILPS with different wind
directions and velocities (a) $\theta=\pi/8$ and $v=1m/s$ (see Visualization
1) (b) $\theta=3\pi/4$ and $v=5m/s$ (see Visualization 2). The direction and
the length of the arrows represent the wind direction and velocity
respectively.

###  3.2 Adaptive optics system

####  3.2.1 Perturbed phase correction

A well-developed AO system comprises three primary assignments: wavefront
measurement, reconstruction, and beam stabilization. The slope magnitude of
beacon light is measured by a Hartmann wavefront sensor, and subsequently used
to invert the voltage signal, which is used to generate the phase distribution
of a deformable mirror, through a reconstructor. For a closed-loop AO system,
the estimated phase is also fed back to compensate the slope magnitude[93],
the combinations of which can continuously improve the phase reconstruction
accuracy.

Since the phase generated by AO system is only an approximation of the
perturbed phase of beacon light, the correction capability of entire system
can be commonly measured by the number of coefficient that is calculated from
the spectral decomposition of beacon light using Zernike polynomial[57, 58].
(e.g., for an ideal AO system, the correction order can reach infinity because
the perturbed phase of beacon light can be expanded to a superposition of an
infinite number of Zernike polynomial with different orders, which is usually
considered as an upper boundary for the performance enhancement of AO). For a
realistic AO system with $N$-th order correction capability, the phase
estimated by a deformable mirror can be expressed as

$\widetilde{\varphi}^{\left(i\right)}\left(r,\phi,z\right)=\sum_{n=1}^{N}a_{n}^{\left(i\right)}\left(z\right)Z_{n}\left(\frac{r}{R},\phi\right),$
(10)

where $\widetilde{\varphi}^{\left(i\right)}\left(r,\phi,z\right)$ is the
estimated phase of beacon light for the $i$-th realization of turbulence and
$Z_{n}\left(\cdot\right)$ is the Zernike polynomial with $n$-th order. The
coefficients $a_{n}^{\left(i\right)}\left(z\right)$ are given by the overlap
integral

$a_{n}^{\left(i\right)}\left(z\right)=\int_{0}^{R}\int_{0}^{2\pi}\varphi^{\left(i\right)}\left(r,\phi,z\right)Z_{n}\left(\frac{r}{R},\phi\right)rdrd\phi,$
(11)

where $\varphi^{\left(i\right)}\left(r,\phi,z\right)$ represents the perturbed
phase of beacon light. After AO correction, the estimated phase will imprint
on the single OAM photon state. We express the action of AO by unitary
operator
$\widehat{U}_{AO}^{\left(i\right)}\equiv\exp\left\\{-i\widetilde{\varphi}^{\left(i\right)}\left(r,\phi,z\right)\right\\}$
with ket-bra notation[60, 61]

$\left|\widetilde{\psi}_{l_{0}}^{\left(i\right)}\right\rangle=\widehat{U}_{AO}^{\left(i\right)}\widehat{U}_{turb}^{\left(i\right)}\left|l_{0}\right\rangle,$
(12)

Finally, we emphasize that, in our AO-aided OAM-based QKD scheme, the signal
part of incident modes consists of photonic OAM states and their Fourier
conjugate angular states (i.e., ANG states). Besides, we employ the platform
beam as a probe state to detect atmospheric turbulence[94]. Our simulations
ensure that two beams are emitted simultaneously and propagated coaxially to
provide a better correction for the signal part.

####  3.2.2 Wrapped cuts elimination

A realistic AO system has been proven to be ineffective at compensating for
branch points that occur at places of zero amplitude in an optical field[95].
These unavoidable phase cuts[64] degrade the performance of OAM-based QKD due
to the inability of a continuous surface deformable mirror to reconstruct the
discontinuous phase[66, 67, 68]. Besides, some previously correlated
researches have demonstrated that branch points begin to appear when the Rytov
variance exceeds $0.1$[96, 97]. Reassuringly, this adverse effect is not
completely insurmountable. Inspired by Refs. [98], we employ the phase
unwrapping algorithm proposed in Ref. [99] to enhance the performance of AO
correction in this paper.

The main idea of our enhanced AO scheme is that we reconstruct the perturbed
phase using the real-time slope information of beacon light and then unwrap
the discontinuous phase by an effective and implementable algorithm;
subsequently, the unwrapped perturbed phase is used to generate the voltage
signal of a deformable mirror. Generally, a standard unwrapping operation on
the perturbed phase $\varphi^{\left(i\right)}\left(r,\phi,z\right)$ can be
accomplished by adding an integer multiple of $2\pi$ at each pixel of wrapped
phase, which is mathematically expressed as[100]

$\Phi^{\left(i\right)}\left(r,\phi,z\right)=\left\\{\begin{tabular}[]{cc}$\varphi^{\left(i\right)}\left(r,\phi,z\right)+2\pi
k\left(r,\phi,z\right)$&$\left(r,\phi\right)\in G$\\\
$\varphi^{\left(i\right)}\left(r,\phi,z\right)$&$\left(r,\phi\right)\notin
G$\end{tabular}\right.,$ (13)

where $\Phi^{\left(i\right)}\left(r,\phi,z\right)$ is the unwrapped perturbed
phase, and $k\left(r,\phi,z\right)$ is the integer that needs to be solved,
regarded as a multi-class classification problem, $G$ stands for the set of
minimum neighborhood containing the wrapped cuts. Considering the perturbed
phase varies with a period of $2\pi$, we indicate that the above operation is
physically feasible. After implementing phase unwrapping, the decomposition
coefficient is modified and can be re-evaluated by

$A_{n}^{\left(i\right)}\left(z\right)=\int_{0}^{R}\int_{0}^{2\pi}\Phi^{\left(i\right)}\left(r,\phi,z\right)Z_{n}\left(\frac{r}{R},\phi\right)rdrd\phi,$
(14)

which leads to a significant performance enhancement compared to the
previously poor correction.

Figure 3: Phase distribitions for a single OAM state with azimuthal index
$l=3$, for a single atmospheric propagation without correction ((a) and (d)),
with realistic correction ((b) and (e)) and ideal correction ((c) and (f)),
the upper and lower three plots represent the realization in the weak and
strong scintillation regime corresponding to $\sigma_{R}^{2}=0.153$ and
$2.235$, respectively, where $\sigma_{R}^{2}$ represents the scintillation
strength, quantified by the Rytov variance. Phase distributions repaired by
our enhanced AO with (b) $30$-order correction and (e) $50$-order correction.
The colorbar for all phase distribution plots (a)–(f) is the same.

##  4 Results

###  4.1 A single OAM state propagation

To quantify the undesirable impact of atmospheric turbulence on OAM-based QKD
and what effects can be obtained through partial correction by AO, the
evolution of a single OAM state that is perturbed while traveling across the
turbulent channel and repaired by a specific-order AO are investigated. In
Fig. 3, we plot the phase distributions of a state $l_{0}=3$ for a single
realization of turbulence with no correction (Fig. 3(a) and 3(d)), realistic
correction[101] (Fig. 3(b) and 3(e)) and ideal correction (Fig. 3(c) and
3(f)), where the upper and lower three diagrams represent the results realized
in the weak and strong scintillation regime, respectively (see more details
about the parameter settings in Ref. [102]).

Fig. 3(a) shows that the phase distribution of OAM state is slightly distorted
after propagation. However, this circumstance becomes more severe in the
strong scintillation regime (Fig. 3(d)), which will lead to a completely
disrupted structure. Notably, we observe in Fig. 3(a) that the initial vortex
splits into three individual vortices, accompanied by the vortex-antivortex
pairs regeneration at places of central point of initial one. Besides,
turbulence also leads to a longer discontinuous cut in the perturbed phase
distribution (For more detailed explanations, we refer the reader to Refs.
[64, 65]). To eliminate the phase discontinuities, a realistic AO is employed
to repair the perturbed phase distribution, as shown in Fig. 3(b) and 3(e)
((b) $30$-order realization (e) $50$-order realization). It is highlighted
that, in the weak scintillation regime, the longest phase cut that occurs in
the perturbed phase is completely eliminated, making the unwrapped corrected
phase is more approach to the unperturbed one (see Fig. 3(b)). On the
contrary, in the strong scintillation regime, the corrected phase remains
almost unchanged compared to the perturbed one (see Fig. 3(e)). Fortunately,
such circumstance is significantly alleviated in ideal correction (see Fig.
3(f)).

The poor correction is likely because the length of branch cuts becomes a
crucial factor in affecting AO correction effectiveness. In other words, in
the weak scintillation regime, only a few branch cuts are generated such that
these can be ignored when we implement realistic AO correction (see more
discussions in Ref. [45]). However, although we eliminate all wrapped cuts in
the strong scintillation regime, the accumulation of branch cuts will destroy
the phase distribution of the beacon light eventually.

In order to evaluate the correction effect of AO intuitively, the crosstalk
probability distributions (including the radial-mode scrambling) averaged over
$500$ realizations of turbulence are presented in Fig. 4. We observe that, in
the weak scintillation regime, our enhanced AO scheme effectively restores the
power into the initial OAM state even in the presence of phase cuts.
Conversely, in the strong scintillation regime, we note that our enhanced AO
loses its advantage to mitigate turbulence-induced crosstalk with the increase
of branch points. Furthermore, we also find the well-known result[61, 103,
104] that the spectral broadening peaks around the incident state. In the
strong scintillation regime, the crosstalk distribution has two peaks around
$l_{0}=3$ and $l_{0}=-3$ (For more discussions about the bimodal distribution,
we refer the reader to Ref. [61]), which results in a smaller average OAM of
the received state compared to that of the incident one[43].

Figure 4: Crosstalk probability distributions, including radial-mode
scrambling, averaged over $500$ realizations of turbulence with azimuthal
index $l=3$, for different degrees of correction (a), (d) no AO; (b), (e)
realistic AO; (c), (f) ideal AO, the upper and lower three plots represent the
realization in the weak and strong scintillation regime, respectively. All
parameter settings are same as in Fig. 3.

###  4.2 Performance of OAM-based QKD across turbulence

Based on the above analysis for the propagation characteristics of a single
OAM state, we now employ our enhanced AO to improve the quality of OAM-based
QKD (without (a) and with (b) increasing the mode spacing in encoding
subspace). Fig. 5 illustrates the variation curves of average QBER without,
with realistic and with ideal AO correction recorded during half an hour for
5-dimensional OAM-based QKD system. we set the correction capability of
realistic AO to $30$-order and $C_{n}^{2}$ to $2.2\times 10^{-15}m^{-2/3}$
during this simulation, where $C_{n}^{2}$ is the refractive index structure
constant (see more detailed parameter settings in Ref. [102] and reasons in
Ref. [105]).

Figure 5: Average QBERs without, with realistic and with ideal AO correction
recorded during half an hour for $5$-dimensional OAM-based QKD system. Panel
(a) and (b) are realized without and with increasing mode spacing in encoding
subspace respectively. The inset is a QBER frequency histogram counted from
the variation curves of the average QBER. The dashed, dash-dotted and dotted
lines in (a), (b) denote the mean value of the average QBER without ((a)
$\overline{Q}=41.9\%$ (b) $\overline{Q}=33.1\%$), with realistic ((a)
$\overline{Q}=11.1\%$ (b) $\overline{Q}=6.9\%$) and with ideal ((a)
$\overline{Q}=2.3\%$ (b) $\overline{Q}=0.9\%$) correction respectively, where
$\overline{Q}\equiv\frac{1}{M}\sum_{i=1}^{M}Q^{\left(i\right)}$, averaged over
$M$ realizations of turbulence. The solid line represents the tolerable QBER
threshold of $5$-dimensional OAM-based QKD. This simulation allows the
correction capability of realistic AO to $30$-order and the turbulence level
to $C_{n}^{2}=2.2\times 10^{-15}m^{-2/3}$. Figure 6: Measured crosstalk
matrices of the OAM basis (a), (d) without, (b), (e) with realistic, and (c),
(f) with ideal correction. The upper and lower three plots represent the
realization without and with increasing mode spacing respectively. All
parameter settings are same as in Fig. 5.

As illustrated in Fig. 5(a), we find that, without increasing mode spacing
(i.e., using $\left|-2\right\rangle$, $\left|-1\right\rangle$,
$\left|0\right\rangle$, $\left|1\right\rangle$ and $\left|2\right\rangle$ for
encoding), the average QBER can be mitigated from $41.9\%$ with a standard
deviation of $6.14\%$ to $11.1\%$ with a standard deviation of $2.93\%$
through the realistic AO correction. Notably, for the ideal circumstance, the
average QBER is reduced by $39.63\%$. With the help of mode spacing increasing
(i.e., using $\left|-4\right\rangle$, $\left|-2\right\rangle$,
$\left|0\right\rangle$, $\left|2\right\rangle$ and $\left|4\right\rangle$ for
encoding, as shown in Fig. 5(b)), we can alleviate the average QBER from
$41.9\%$ to $33.1\%$ without correction, $11.1\%$ to $6.9\%$ with realistic
correction and $2.3\%$ to $0.9\%$ with ideal correction, which indicates that
the combination of two strategies can lead to a significant reduction of
average QBER caused by atmospheric turbulence and rebuild a secure channel
between two communication parties. In Fig. 6, we also evaluate the measured
crosstalk matrices of perturbed OAM basis under the same settings of Fig. 5
without and with AO correction respectively. More detailed discussions about
the crosstalk matrices are similar to the analysis presented in Fig. 5.

However, will AO correction work so well in the real-world? Unfortunately,
when we implement our enhanced AO-aided OAM-based QKD experiment, the
instrument inevitably generates some intrinsic noises (such as dark
counts[29], shot noise[73], afterpulsing effect[87], time delay effect[54, 69,
70], to name just a few); besides, the two MUBs will also encounter different
degrees of loss due to the receiving aperture[73], atmospheric aerosols and
dust particles[106] during the propagation (the last two factors are not
considered in this contribution, see reasons in Ref. [107]). The accumulation
of these realistic noises can remarkably lead to an increased average QBER and
seriously degrade the performance of realistic AO system. Hence, we have to
emphasize that the reason why AO correction can achieve the high-quality
enhancement in the above simulation is that the above noise contributions are
not considered, which will be mentioned in subsection 5.

###  4.3 Under arbitrary turbulence strengths

In this section, we describe how the average QBER and the secret key rate of
OAM-based QKD change with respect to the atmospheric coherence length $r_{0}$
under different dimensions of encoding in Fig. 7, where each column from left
to right represents the results without (Fig. 7(a) and (d)), with realistic
(Fig. 7(b) and (e)) and with ideal (Fig. 7(c) and (f)) AO correction
respectively. The dashed vertical lines located at $r_{0}\approx 0.017m$ in
all six plots correspond to the onset of weak scintillation. The atmospheric
and optical parameter settings adopted in this subsection are listed as
follows: $v=1m/s$, $\theta=\pi/2$, $z=1000m$, $\omega_{0}=0.03m$ and
$\lambda=632nm$.

Figure 7: Average QBER (a)–(c) and secret key rate (d)–(f) of OAM-based QKD as
a function of $r_{0}$ with different dimensions of encoding. Different degrees
of correction are considered: (a), (d) no AO; (b), (e) realistic AO; (c), (f)
ideal AO, averaged over $300$ realizations of turbulence. The dashed vertical
lines at $r_{0}\approx 0.017m$ in all six plots correspond to
$\sigma_{R}^{2}=1$. The error bars represent the standard error. The
correction capability of AO and mode spacing is set to $30$-order and $1$ in
both cases.

From the results presented in Fig. 7(a), we observe that the average QBER
gradually decreases as the turbulence becomes weaker. For a higher dimensional
OAM-based QKD, the average QBER is overall superior to the lower dimensional
encoding in the strong scintillation regime, which is likely explained by the
fact that the OAM eigenstates with a higher azimuthal index have a larger beam
size so that they are more susceptible to atmospheric turbulence. In the weak
scintillation regime, all these differences between different curves become
smaller with decreasing turbulence strength. Besides, as illustrated in Fig.
7(d), we see that no positive key rate can be obtained by OAM-based QKD in all
considered dimensions under moderate to strong turbulence strengths (i.e.,
$r_{0}$ ranges from $0.01m$ to $0.06m$), which causes both parties cannot
build a secure communication link. Generally, we expect to improve the system
security and information capacity of OAM-based QKD when we employ high
dimensional encoding. However, we find in Fig. 7(d) that when $r_{0}\approx
0.12m$, a lower secret key rate will be obtained despite a higher dimensional
encoding subspace is used (i.e., a more secure communication link guaranteed
by a higher dimensional OAM-based QKD is destroyed by atmospheric turbulence).
Fortunately, high dimensional OAM-based QKD will reveal its unique advantages
as the turbulence strength gradually decreases. Therefore, it should be noted
that, for a given turbulence strength, selecting an appropriate dimension for
encoding is especially critical to acquire a better performance of OAM-based
QKD (e.g., we choose $d=3$ for $r_{0}\approx 0.12m$ and $d=7$ for
$r_{0}\approx 0.25m$).

Figure 8: Average QBER (a)–(c) and secret key rate (d)–(f) of OAM-based QKD as
a function of $r_{0}$ with different dimensions of encoding. Different degrees
of correction are considered: (a), (d) no AO; (b), (e) realistic AO; (c), (f)
ideal AO, averaged over $300$ realizations of turbulence. The dashed vertical
lines at $r_{0}\approx 0.017m$ in all six plots correspond to
$\sigma_{R}^{2}=1$. The error bars represent the standard error. The
correction capability of AO and mode spacing is set to $30$-order and $2$ in
both cases.

The central and last columns in Fig. 7 illustrate the QKD’s performance
achieved with realistic and ideal AO correction. Based on the comparison
between Fig. 7(a) and (b), we see that, in the weak scintillation regime, the
average QBER significantly decreases as the turbulence becomes weaker, which
implies that our enhanced AO is effective for improving the quality of OAM-
based QKD. Specifically, we notice that the realistic AO can achieve the best
performance enhancement under moderate turbulence strengths (e.g., when
$r_{0}=0.044m$ and $d=5$, the average QBER decreases from $44\%$ to $14\%$ and
the secret key improves from no positive keys to $0.605$ bits/photon). On the
other hand, we re-evaluate in Fig. 7(e) how the secret key rate changes with
respect to $r_{0}$ under the realistic AO correction. Compared to Fig. 7(d),
we observe that the realistic AO correction can lead to a positive key rate
when the turbulence strength is moderate, which substantially improves the
utilization of OAM-based QKD. Furthermore, with the help of ideal correction,
we obtain an upper boundary (Fig. 7(c) and (f)) for the performance
enhancement of OAM-based QKD, which leads to a positive key rate in all
considered turbulence strengths (e.g., when $r_{0}=0.02m$ and $d=5$, the
secret key rate improves from no positive keys to $1.04$ bits/photon).

We also enhance the performance of OAM-based QKD through increasing mode
spacing in encoding subspace. Comparing the results obtained between the two
strategies (Fig. 7 and 8), we note that, in the strong scintillation regime,
when the mode spacing becomes $2$, the average QBER of OAM-based QKD
counterintuitively increases compared to the successive encoding (see the
comparison between Fig. 7(a) and 8(a), e.g., when $r_{0}=0.01m$, the average
QBER without AO correction improves from $75.32\%$ to $80.32\%$ for $d=7$,
$68.98\%$ to $73.95\%$ for $d=5$ and $56.49\%$ to $60.88\%$ for $d=3$, more
explanations are presented in Ref. [38]). However, such performance
degradation does not appear when we employ the ideal correction (see the
comparison between Fig. 7(c) and 8(c)), which is a compromise result between
the degradation and correction. Besides, in the weak scintillation regime, we
conclude that increasing the mode spacing can partially improve the quality of
OAM-based QKD, especially with the help of AO (e.g., when $r_{0}=0.15m$ and
$d=5$, the secret key rate improves from $1.85$ bits/photon to $2.1$
bits/photon with realistic AO correction and from $2.04$ bits/photon to $2.27$
bits/photon with ideal AO correction).

###  4.4 Under arbitrary correction orders

Since the correction capability of realistic AO system is commonly determined
by its order, we evaluate how the performance of an AO-aided OAM-based QKD
changes with respect to the correction order under different dimensions of
encoding in Fig. 9. We set $C_{n}^{2}=1.01\times 10^{-14}m^{-2/3}$ during the
simulation. It can be seen that the quality of OAM-based QKD improves
significantly as the correction order of AO increases. However, such
significant performance enhancement is only in the lower order correction
range. The primary reason because the turbulence-induced aberrations is almost
entirely concentrated in the low-frequency part. On the other hand, since we
set the turbulence strength in the moderate to strong range, we notice that
when the correction order is larger than $100$-order, the quality of OAM-based
QKD still improves somewhat as the correction order increases. Generally, we
anticipate that the quality of OAM-based QKD might not be significantly
enhanced with the increase of the correction order of AO under the weak
turbulence strengths. In Fig. 9(b) and 9(d), we also present the results of
increasing mode spacing in encoding subspace. We observe that only a minor
performance enhancement appears compared to the successive encoding.

Figure 9: Average QBER (a), (b) and secret key rate (c), (d) of OAM-based QKD
as a function of correction order with different dimensions of encoding. Panel
(a), (c) and panel (b), (d) represent the results realized without and with
increasing mode spacing respectively. All calculations in four plots are 300
realizations of turbulence. The error bars represent the standard error. The
turbulence level in four plots is $C_{n}^{2}=1.01\times 10^{-14}m^{-2/3}$.

Another feature can be seen in Fig. 9(c) and (d) is that when we increase the
correction order of AO and mode spacing in encoding subspace, we can recover
the advantage of high-dimensional encoding to some extent, but choosing
$3$-dimensional OAM states for encoding is still the best choice under this
turbulence strength.

###  4.5 Realistic considerations

Now, we evaluate the performance enhancement achieved by our enhanced AO with
more realistic considerations, including the effects of loss from a limited
sized receiving aperture and time delay between wavefront measurement and
correction.

####  4.5.1 Loss

The state-dependent loss[73, 109] caused by the diffraction effects
significantly degrades the performance of OAM-based QKD in long-distance
propagation. In the two MUBs, an OAM eigenstate with a larger azimuthal number
suffers from more significant loss and acquires more propagation phase[83] due
to the limited size of receiving aperture, leading to a reduced quality of QKD
even in the absence of turbulence. In order to evaluate the impact of state-
dependent loss on the performance of OAM-based QKD, we re-examine how the
average QBER and the secret key rate change with respect to $r_{0}$ under
different sizes of receiving aperture. Here, we only consider the performance
degradation in 5-dimensional OAM-based QKD. the other parameter settings
adopted during the simulation is the same as that in subsection 3.

Figure 10: Average QBER (a)–(c) and secret key rate (d)–(f) of $5$-dimensional
OAM-based QKD as a function of $r_{0}$ with different sizes of receiving
aperture. Different degrees of correction are considered: (a), (d) no AO; (b),
(e) realistic AO; (c), (f) ideal AO, averaged over $300$ realizations of
turbulence. The dashed vertical lines at $r_{0}\approx 0.017m$ in all six
plots correspond to $\sigma_{R}^{2}=1$. The error bars represent the standard
error. All other parameter settings are same as in Fig. 7.

As illustrated in Fig. 10(a), we observe that when the channel is under strong
turbulence strengths and without AO correction, reducing the receiving
aperture can surprisingly mitigate the impact of atmospheric turbulence to
some extent (e.g., when $r_{0}=0.034m$, the average QBER decreases from
$53.4\%$ to $44.98\%$), which is somewhat counterintuitive. The main reason
may be that, under these turbulence strengths, the smaller size of the
aperture is similar to a low-pass filter[110], a large amount of turbulence-
induced vortex-antivortex pairs may wander outside the receiving aperture and
result in a decrease of average QBER[111, 112]. Conversely, in the presence of
AO correction, we notice that decreasing the size of receiving aperture will
lead to a significant increase to average QBER (Fig. 10(b) and (c), e.g., when
$r_{0}=0.01m$, the average QBER increases from $23.77\%$ to $35.93\%$ with
ideal AO correction) and decrease to secret key rate (Fig. 10(e) and (f),
e.g., when $r_{0}=0.31m$, the secret key rate decreases from $2.89$
bits/photon to $1.32$ bits/photon with ideal AO correction). Such an
observation is likely because the aperture loss becomes the primary source of
bit error for OAM-based QKD after implementing AO correction.

####  4.5.2 Time delay

In above simulation, we investigate the quality enhancement of OAM-based QKD
based on the assumption that AO correction is implemented in real time.
However, since our enhanced AO includes the procedure of reconstructing and
unwrapping the phase of beacon light, the impact of time delay effect between
wavefront measurement and correction on OAM-based QKD cannot be ignored, which
can also significantly affect the performance of OAM-based QKD. It is well-
known that real-time implementation can not be realized except the Greenwood
frequency is small enough for the bandwidth of AO system[113]. In order to re-
evaluate the impact of performance degradation caused by the time delay effect
on OAM-based QKD, we set the bandwidth of AO as one-fifth of the sampling
frequency of a Hartmann camera[114] (i.e., if we assume the camera sample
frequency equals to $1KHz$, then $f_{AO}=200Hz$), and change the Greenwood
frequency $f_{G}$ by adjusting the wind velocity through the following
relationship[54]

$f_{G}=\frac{0.43v}{r_{0}}.$ (15)

Figure 11: Average QBER of $5$-dimensional OAM-based QKD as a function of
$r_{0}$ with different wind velocities, averaged over $300$ realizations of
turbulence. The horizontally dotted line represents the ten-times time delay
of AO (defined as $\tau_{AO}\equiv 1/f_{AO}$) if we assume the bandwidth of AO
is $200Hz$ (i.e., $f_{AO}=200Hz$). The dash-dotted lines denote the Greenwood
time $\tau_{G}$ according to Eq. (15), Refs. [38, 58] indicate that the
crossing point between the dash-dotted lines and the dotted line represents
the bandwidth of AO is large enough to provide a comprehensive correction. The
dashed lines represent the value of $r_{0}$ corresponding to the intersection
of $\tau_{G}$ and $10\tau_{AO}$ for different wind velocities. The correction
capability of AO is same as in Fig. 7. The error bars represent the standard
error.

In Fig. 11, we consider how the average QBER changes with respect to $r_{0}$
under different wind velocities. We observe that when the wind velocity is
small (i.e., $v=1m/s$), the evolution of average QBER almost coincides with
the results obtained without time delay. On the contrary, with the increase of
wind velocity, the average QBER starts to deviate from the no time delay
counterparts, leading to the performance degradation of OAM-based QKD (e.g.,
when $r_{0}=0.026m$, the average QBER increases from $28.34\%$ to $40.54\%$
for $v=5m/s$, from $28.34\%$ to $49.78\%$ for $v=10m/s$). Besides, we also
notice that when the turbulence becomes weaker, the deviation together
inclines to disappear. To explain above phenomena, Fig. 11 is horizontally
divided into two parts[115]. We observe that when $v=5m/s$ and the Greenwood
time (defined as $\tau_{G}\equiv 1/f_{G}$) passes this threshold, the
bandwidth of AO is large enough to provide a comprehensive correction (see
more discussions in Ref. [38]). However, these conclusions are not established
for $v=10m/s$, which implies that, in the large wind velocity regime, the
bandwidth of AO needs to be enhanced to satisfy the fast time-varying
wavefront sampling and compensation. Finally, by comparing to the results
obtained without AO correction, we see that AO remains able to mitigate the
performance degradation even though it works in the large wind velocity
environment.

##  5 Discussion and conclusion

In this paper, we employ the ILPS method to simulate the performance
degradation of OAM-based QKD under arbitrary turbulence strengths. The main
idea behind ILPS is to use the partial data from previous phase screen to
generate the results of next one, which builds a connection between two
generated phase screens. An advantage of this design is that the wind velocity
of turbulence can be simulated by adjusting the amount of motion in each
iteration, which provides a feasible way to simulate the dynamical process of
incident light propagation and how AO dynamically corrects the turbulence-
induced aberrations in real time. However, we have to highlight the
shortcoming of ILPS, which uses the spatial correlation of phase screen
between two iterations to replace the temporal correlation of turbulence
variation (e.g., the variation of wind velocity increases the Greenwood
frequency of turbulence[54, 70], however, ILPS does not perform well in the
face of turbulence variation in a windless environment), a topic needs to be
addressed in the future.

Considering the inability of traditional AO while encountering phase cuts[66,
67, 68], we demonstrate the feasibility of AO that includes the wrapped cuts
elimination, which significantly alleviates the impact of atmospheric
turbulence on OAM-based QKD. We show that, in the weak scintillation regime,
the employment of realistic AO for correcting the turbulence-induced
aberrations can lead to a $30\%$ performance enhancement approximately.
However, in the strong scintillation regime, we observe a rapid reduction of
the performance of AO correction. Furthermore, we suggest that if we wish to
obtain a high-quality performance enhancement, an advanced AO system is
necessary for the realistic experiments[38].

Since the additional noise contributions (such as state-dependent loss, time
delay effect of AO, to name just a few) degrade the high-quality enhancement
achieved by AO, we re-evaluate the correction effect of AO and conclude that
AO can still rebuild a secure communication channel and recover the high
information capacity of OAM-based QKD, even in the increased loss and wind
velocity environment. However, for the real-time correction consideration, we
still need to enhance the bandwidth of AO when the wind velocity becomes
larger.

Finally, it should be noted that although our enhanced AO includes the
procedure for eliminating wrapped cuts, the branch cuts caused by turbulence-
induced aberrations cannot be entirely negated by traditional AO techniques.
Further research on minimizing the length of branch cuts that occur in the
perturbed phase may prove fruitful.

## Funding

Anhui Provincial Natural Science Foundation (1908085QA37); National Natural
Science Foundation of China (11904369); State Key Laboratory of Pulsed Power
Laser Technology Supported by Open Research Fund of State Key Laboratory of
Pulsed Power Laser Technology (2019ZR07).

## Acknowledgments

The authors are very grateful to the reviewers for their valuable comments. Z.
T. thanks Prof. Ruizhong Rao for his careful reading and insightful
suggestions.

## Disclosures

The authors declare no conflicts of interest.

## References

  * [1] C. H. Bennett and G. Brassard, "Quantum cryptography: public key distribution and coin tossing," Proceedings of the IEEE International, Conference on Computers, Systems and Signal Processing pp. 175–179 (1984).
  * [2] C. H. Bennett, "Quantum cryptography using any two nonorthogonal states," Phys. Rev. Lett. 68(21), 3121–3125 (1992).
  * [3] D. Bruß, "Optimal eavesdropping in quantum cryptography with six states," Phys. Rev. Lett. 81(15), 3018–3021 (1998).
  * [4] V. Scarani, A. Acín, G. Ribordy, and N. Gisin, "Quantum cryptography protocols robust against photon number splitting attacks for weak laser pulse implementations," Phys. Rev. Lett. 92(5), 057901 (2004).
  * [5] A. K. Ekert, "Quantum cryptography based on Bell’s theorem," Phys. Rev. Lett. 67(6), 661–663 (1991).
  * [6] C. H. Bennett, G. Bennett, N. D. Mermin, "Quantum cryptography without Bell’s theorem," Phys. Rev. Lett. 68(5), 557–559 (1992).
  * [7] H. K. Lo and H. F. Chau, "Unconditional security of quantum key distribution over arbitrarily long distances," Science 283(5410), 2050–2056 (1999).
  * [8] P. Shor and J. Preskill, "Simple proof of security of the BB84 quantum key distribution protocol," Phys. Rev. Lett. 85(2), 441 (2000).
  * [9] B. Kraus, N. Gisin, and R. Renner, "Lower and upper bounds on the secret-key rate for quantum key distribution protocols using one-way classical communication," Phys. Rev. Lett. 95(8), 080501 (2005).
  * [10] R. Renner, N. Gisin, and B. Kraus, "Information-theoretic security proof for quantum-key-distribution protocols," Phys. Rev. A 72(1), 012332 (2005).
  * [11] V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, "The security of practical quantum key distribution," Rev. Mod. Phys. 81(3), 1301–1350 (2009).
  * [12] D. Gottesman, H. K. Lo, N. Lütkenhaus, and J. Preskill, "Security of quantum key distribution with imperfect devices," Quantum Inf. Comput. 4(5), 325–360 (2004).
  * [13] W. K. Wootters and W. H. Zurek, "A single quantum cannot be cloned," Nature 299(5886), 802–803 (1982).
  * [14] N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, "Quantum cryptography," Rev. Mod. Phys. 74(1), 145–195 (2002).
  * [15] C. Gobby, Z. L. Yuan, and A. J. Shields, "Quantum key distribution over 122 km of standard telecom fiber," Appl. Phys. Lett. 84(19), 3762–3764 (2004).
  * [16] A. Poppe, A. Fedrizzi, R. Ursin, H. R. Böhm, T. Lorünser, O. Maurhardt, M. Peev, M. Suda, C. Kurtsiefer, H. Weinfurter, T. Jennewein, and A. Zeilinger, "Practical quantum key distribution with polarization entangled photons," Opt. Express 12(16), 3865–3871 (2004).
  * [17] T. Schmitt-Manderbach, H. Weier, M. Fürst, R. Ursin, F. Tiefenbacher, T. Scheidl, J. Perdigues, Z. Sodnik, C. Kurtsiefer, J. G. Rarity, A. Zeilinger, and H. Weinfurter, "Experimental demonstration of free-space decoy-state quantum key distribution over 144 km," Phys. Rev. Lett. 98(1), 010504 (2007).
  * [18] M. Sasaki, M. Fujiwara, H. Ishizuka, W. Klaus, K. Wakui, M. Takeoka, S. Miki, T. Yamashita, Z. Wang, A. Tanaka, K. Yoshino, Y. Nambu, S. Takahashi, A. Tajima, A. Tomita, T. Domeki, T. Hasegawa, Y. Sakai, H. Kobayashi, T. Asai, K. Shimizu, T. Tokura, T. Tsurumaru, M. Matsui, T. Honjo, K. Tamaki, H. Takesue, Y. Tokura, J. F. Dynes, A. R. Dixon, A. W. Sharpe, Z. L. Yuan, A. J. Shields, S. Uchikoga, M. Legré, S. Robyr, P. Trinkler, L. Monat, J. B. Page, G. Ribordy, A. Poppe, A. Allacher, O. Maurhart, T. Länger, M. Peev, and A. Zeilinger, "Field test of quantum key distribution in the Tokyo QKD Network," Opt. Express 19(11), 10387–10409 (2011).
  * [19] J. Yin, Y. Cao, Y.-H. Li, S.-K. Liao, L. Zhang, J.-G. Ren, W.-Q. Cai, W.-Y. Liu, B. Li, H. Dai, G.-B. Li, Q.-M. Lu, Y.-H. Gong, Y. Xu, S.-L. Li, F.-Z. Li, Y.-Y. Yin, Z.-Q. Jiang, M. Li, J.-J. Jia, G. Ren, D. He, Y.-L. Zhou, X.-X. Zhang, N. Wang, X. Chang, Z.-C. Zhu, N.-L. Liu, Y.-A. Chen, C.-Y. Lu, R. Shu, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, "Satellite-based entanglement distribution over 1200 kilometers," Science 356(6343), 1140–1144 (2017).
  * [20] S.-K. Liao, W.-Q. Cai, W.-Y. Liu, L. Zhang, Y. Li, J.-G. Ren, J. Yin, Q. Shen, Y. Cao, Z.-P. Li, F.-Z. Li, X.-W. Chen, L.-H. Sun, J.-J. Jia, J.-C. Wu, X.-J. Jiang, J.-F. Wang, Y.-M. Huang, Q. Wang, Y.-L. Zhou, L. Deng, T. Xi, L. Ma, T. Hu, Q. Zhang, Y.-A. Chen, N.-L. Liu, X.-B. Wang, Z.-C. Zhu, C.-Y. Lu, R. Shu, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, "Satellite-to-ground quantum key distribution," Nature 549(7670), 43–47 (2017).
  * [21] L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 45(11), 8185–8189 (1992).
  * [22] H. Bechmann-Pasquinucci and W. Tittel, "Quantum cryptography using larger alphabets," Phys. Rev. A 61(6), 062308 (2000).
  * [23] H. Bechmann-Pasquinucci and A. Peres, "Quantum cryptography with 3-state systems," Phys. Rev. Lett. 85(15), 3313–3316 (2000).
  * [24] N. J. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, "Security of Quantum Key Distribution Using $d$-Level Systems," Phys. Rev. Lett. 88(12), 127902 (2002).
  * [25] L. Sheridan and V. Scarani, "Security proof for quantum key distribution using qudit systems," Phys. Rev. A 82(3), 030301(R) (2012).
  * [26] S. Grölacher, T. Jennewein, A. Vaziri, G. Weihs, and A. Zeilinger, "Experimental quantum cryptography with qutrits," New J. Phys. 8(5), 75 (2006).
  * [27] M. Malik, M. O’Sullivan, B. Rodenburg, M. Mirhosseini, J. Leach, M. P. J. Lavery, M. J. Padgett, and R. W. Boyd, "Influence of atmospheric turbulence on optical communications using orbital angular momentum for encoding," Opt. Express 20(12), 13195–13200 (2012).
  * [28] V. D’Ambrosio, E. Nagali, S. P. Walborn, L. Aolita, S. Slussarenko, L. Marrucci, and F. Sciarrino, "Complete experimental toolbox for alignment-free quantum communication," Nat. Commun. 3, 961 (2012).
  * [29] M. Mafu, A. Dudley, S. Goyal, D. Giovannini, M. McLaren, M. J. Padgett, T. Konrad, F. Petruccione, N. Lükenhaus, and A. Forbes, "Higher-dimensional orbital-angular-momentum-based quantum key distribution with mutually unbiased bases," Phys. Rev. A 88(3), 032305 (2013).
  * [30] M. Mirhosseini, O. S. Magaňa-Loaiza, M. N. O’Sullivan, B. Rodenburg, M. Malik, M. P. J. Lavery, M. J. Padgett, D. J. Gauthier, and R. W. Boyd, "High-dimensional quantum cryptography with twisted light," New J. Phys. 17(3), 033033 (2015).
  * [31] H. Larocque, J. Gagnon-Bischoff, D. Mortimer, Y. Zhang, F. Bouchard, J. Upham, V. Grillo, R. W. Boyd, and E. Karimi, "Generalized optical angular momentum sorter and its application to high-dimensional quantum cryptography," Opt. Express 25(17), 19832–19843 (2017).
  * [32] B. Ndagano, I. Nape, B. Perez-Garcia, S. Scholes, R. I. Hernandez-Aranda, T. Konrad, M. P. J. Lavery, and A. Forbes, "A deterministic detector for vector vortex states," Sci. Rep. 7(1), 13882–8 (2017).
  * [33] I. Nape, E. Otte, A. Valles, C. Rosales-Guzmán, F. Cardano, C. Denz, and A. Forbes, "Self-healing high-dimensional quantum key distribution using hybrid spin-orbit Bessel states," Opt. Express 26(21), 26946–26960 (2018).
  * [34] F. Bouchard, K. Heshami, D. England, R. Fickler, R. W. Boyd, B. G. Englert, L. L. Sáchez-Soto, and E. Karimi, "Experimental investigation of high-dimensional quantum key distribution protocols with twisted photons," Quantum 2, 111 (2018).
  * [35] F. X. Wang, W. Chen, Z. Q. Yin, S. Wang, G. C. Guo, and Z. F. Han, "Characterizing high-quality high-dimensional quantum key distribution by state mapping between different degrees of freedom," Phys. Rev. Applied 11(2), 024070 (2019).
  * [36] G. Vallone, V. D’Ambrosio, A. Sponselli, S. Slussarenko, L. Marrucci, F. Sciarrino, and P. Villoresi, "Free-space quantum key distribution by rotation-invariant twisted photons," Phys. Rev. Lett. 113(6), 060503 (2014).
  * [37] A. Sit, F. Bouchard, R. Fickler, J. Gagnon-Bischoff, H. Larocque, K. Heshami, D. Elser, C. Peuntinger, K. Günthner, B. Heim, C. Marquardt, G. Leuchs, R. W. Boyd, and E. Karimi, "High-dimensional intracity quantum cryptography with structured photons," Optica 4(9), 1006–1010 (2017).
  * [38] J. Zhao, Y. Zhou, B. Braverman, C. Liu, K. Pang, N. K. Steinhoff, G. A. Tyler, A. E. Willner, and R. W. Boyd, "Performance of real-time adaptive optics compensation in a turbulent channel with high-dimensional spatial-mode encoding," Opt. Express 28(10), 15376–15391 (2020).
  * [39] F. Bouchard, A. Sit, F. Hufnagel, A. Abbas, Y. Zhang, K. Heshami, R. Fickler, C. Marquardt, G. Leuchs, R. W. Boyd, and E. Karimi, "Quantum cryptography with twisted photons through an outdoor underwater channel," Opt. Express 26(17), 22563–22573 (2018).
  * [40] F. Hufnagel, A. Sit, F. Grenapin, F. Bouchard, K. Heshami, D. England, Y. Zhang, B. J. Sussman, R. W. Boyd, G. Leuchs, and E. Karimi, "Characterization of an underwater channel for quantum communications in the Ottawa River," Opt. Express 27(19), 26346–26354 (2019).
  * [41] F. Hufnagel, A. Sit, F. Bouchard, Y. Zhang, D. England, K. Heshami, B. J. Sussman, and E. Karimi, "Investigation of underwater quantum channels in a 30 meter flume tank using structured photons," New J. Phys. 22, 093074 (2020).
  * [42] Y. Zhou, J. Zhao, B. Braverman, K. Pang, R. Zhang, A. E. Willner, Z. Shi, and R. W. Boyd, "Multiprobe time reversal for high-fidelity vortex-mode-division multiplexing over a turbulent free-space link," Phys. Rev. Applied 15(3), 034011 (2021).
  * [43] M. P. J. Lavery, C. Peuntinger, K. Gunthner, P. Banzer, D. Elser, R. W. Boyd, M. J. Padgett, C. Marquardt, and G. Leuchs, "Free-space propagation of high-dimensional structured optical fields in an urban environment," Sci. Adv. 3(10), e1700552 (2017).
  * [44] M. P. J. Lavery, "Vortex instability in turbulent free-space propagation," New J. Phys. 20, 043023 (2018).
  * [45] G. Sorelli, V. N. Shatokhin, and A. Buchleitner, "Photonic orbital angular momentum in turbulence: vortex splitting and adaptive optics," Proc. SPIE 11532, 115320E (2020).
  * [46] C. Paterson, "Atmospheric turbulence and orbital angular momentum of single photons for optical communication," Phys. Rev. Lett. 94(15), 153901 (2005).
  * [47] G. A. Tyler and R. W. Boyd, "Influence of atmospheric turbulence on the propagation of quantum states of light carrying orbital angular momentum," Opt. Lett. 34(2), 142–144 (2009).
  * [48] C. Chen and H. Yang, "Characterizing the radial content of orbital-angular-momentum photonic states impaired by weak-to-strong atmospheric turbulence," Opt. Express 24(17), 19713–19727 (2016).
  * [49] C. M. Mabena and F. S. Roux, "Optical orbital angular momentum under strong scintillation," Phys. Rev. A 99(1), 013828 (2019).
  * [50] B. J. Smith and M. G. Raymer, "Two-photon wave mechanics," Phys. Rev. A 74(6), 062104 (2006).
  * [51] A. H. Ibrahim, F. S. Roux, and T. Konrad, "Parameter dependence in the atmospheric decoherence of modally entangled photon pairs," Phys. Rev. A 90(5), 052115 (2014).
  * [52] N. D. Leonhard, V. N. Shatokhin, and A. Buchleitner, "Universal entanglement decay of photonic-orbital-angular-momentum qubit states in atmospheric turbulence," Phys. Rev. A 91(1), 012345 (2015).
  * [53] Z. Wang, R. Malaney, and B. Burnett, "Satellite-to-Earth quantum key distribution via orbital angular momentum," Phys. Rev. Applied 14(6), 064031 (2020).
  * [54] J. M. Beckers, "Adaptive optics for astronomy: principles, performance, and applications," Annu. Rev. Astron. Astrophys. 31(1), 13–62 (1993).
  * [55] R. R. Parenti, "Adaptive Optics for Astronomy," Lincoln Lab. J. 5, 93–113 (1992).
  * [56] F. Roddier, Adaptive optics in astronomy (Cambridge University, 1999).
  * [57] R. K. Tyson, Principles of Adaptive Optics, 3rd edn (CRC Press, 2011).
  * [58] R. K. Tyson and B. W. Frazier, Field guide to adaptive optics (SPIE, 2012).
  * [59] J. R. Gonzalez Alonso and T. Brun, "Recovering Quantum Information in Orbital Angular Momentum of Photons by Adaptive Optics," arXiv preprint arXiv:1612.02552 (2016).
  * [60] N. Leonhard, G. Sorelli, V. N. Shatokhin, C. Reinlein, and A. Buchleitner, "Protecting the entanglement of twisted photons by adaptive optics," Phys. Rev. A 97(1), 012321 (2018).
  * [61] G. Sorelli, N. Leonhard, V. N. Shatokhin, C. Reinlein, and A. Buchleitner, "Entanglement protection of high-dimensional states by adaptive optics," New J. Phys. 21(1), 023003 (2019).
  * [62] G. I. Taylor, "Statistical theory of turbulence," Proc. Roy. Soc. London A 151(873), 421–444 (1935).
  * [63] F. Asséat, R. Wilson, and E. Gendron, "Method for simulating infinitely long and non stationary phase screens with optimized memory storage" Opt. Express 14(3), 988–999 (2006).
  * [64] D. L. Fried and J. L. Vaughn, "Branch cuts in the phase function," Appl. Opt. 31(15), 2865–2882 (1992).
  * [65] Z. Tao, Y. Ren, A. Abdukirim, S. Liu, and R. Rao, "Physical meaning of the deviation scale under arbitrary turbulent strengths of optical orbital angular momentum," arXiv preprint arXiv:2102.01891 (2021).
  * [66] D. L. Fried, "Branch point problem in adaptive optics," J. Opt. Soc. Am. A 15(10), 2759–2768 (1998).
  * [67] D. L. Fried, "Adaptive optics wave function reconstruction and phase unwrapping when branch points are present," Opt. Commun. 200, 43–72 (2001).
  * [68] D. L. Fried, "Nature of the branch point problem in adaptive optics," Proc. SPIE 3381, 38–46 (1998).
  * [69] D. L. Fried, "Time-delay-induced mean-square error in adaptive optics," J. Opt. Soc. Am. A 7(7), 1224-1225 (1990).
  * [70] J. Li, Z. Zhang, J. Gao, J. Sun, and W. Chen, "Bandwidth of adaptive optics system in atmospheric coherent laser communication," Opt. Commun. 359, 254–260 (2016).
  * [71] It is worth highlighting that there is no essential difference between a conventional Hartmann wavefront sensor and a Shack-Hartmann wavefront sensor. The latter only replaces the pinhole of a conventional wavefront sensor with a microlens array (i.e., the former is pinhole-imaging, while the latter is convex-lens-imaging).
  * [72] It is noted that LG modes chosen here for encoding is not the best choice, but a compromise consideration. Actually, in order to reduce the loss due to mode dependent diffraction, we commonly choose the eigenstate of $H$ (i.e., the mode with the lowest loss for free-space propagation, for more details about the eigenstate and definition of $H$, we refer the reader to Refs. [73, 74]) as a probe state for OAM encoding[75], which is consistent with the mathematical solution of the communication modes proposed in [76] under the circular aperture. When the Fresnel number becomes larger, the eigenstates of $H$ degenerate to LG modes. Moreover, the communication implemented by LG modes can achieve a comparable performance compared to the eigenstates of $H$ under weak turbulence strengths[77].
  * [73] J. Zhao, M. Mirhosseini, B. Braverman, Y. Zhou, S. M. Hashemi Rafsanjani, Y. Ren, N. K. Steinhoff, G. A. Tyler, A. E. Willner and R. W. Boyd, "Performance analysis of $d$-dimensional quantum cryptography under state-dependent diffraction," Phys. Rev. A 100(3), 032319 (2019).
  * [74] G. A. Tyler, "Spatial bandwidth considerations for optical communication through a free space propagation link," Opt. Lett. 36(23), 4650–4652 (2011).
  * [75] J. Zhao, The Institute of Optics, University of Rochester, 2020 Rochester, N. Y., 14627 (personal communication, 2020).
  * [76] D. A. B. Miller, "Waves, modes, communications, and optics: a tutorial," Adv. Opt. Photonics. 11(3), 679–825 (2019).
  * [77] L. Borcea, J. Garnier, and K. Sølna, "Multimode communication through the turbulent atmosphere," J. Opt. Soc. Am. A 37(5), 720–730 (2020).
  * [78] E. Karimi and E. Santamato, "Radial coherent and intelligent states of paraxial wave equation," Opt. Lett. 37(13), 2484–2486 (2012).
  * [79] E. Karimi, R. W. Boyd, P. de la Hoz, H. de Guise, J. Řeháček, Z. Hradil, A. Aiello, G. Leuchs, and L. L. Sánchez-Soto, "Radial quantum number of Laguerre-Gauss modes," Phys. Rev. A 89(6), 063813 (2014).
  * [80] E. Karimi, D. Giovannini, E. Bolduc, N. Bent, F. M. Miatto, M. J. Padgett, and R. W. Boyd, "Exploring the quantum nature of the radial degree of freedom of a photon via Hong-Ou-Mandel interference," Phys. Rev. A 89(1), 013829 (2014).
  * [81] W. N. Plick and M. Krenn, "Physical meaning of the radial index of Laguerre-Gauss beams," Phys. Rev. A 92(6), 063841 (2015).
  * [82] J. D. Schmidt, Numerical Simulation of Optical Wave Propagation with Examples in MATLAB (SPIE, 2010).
  * [83] M. J. Padgett, F. M. Miatto, M. P. J. Lavery, A. Zeilinger, and R. W. Boyd, "Divergence of an orbital-angular-momentum-carrying beam upon propagation," New J. Phys. 17(2), 023011 (2015).
  * [84] M. Charnotskii, "Extended Huygens–Fresnel principle and optical waves propagation in turbulence: discussion," J. Opt. Soc. Am. A 32(7), 1357–1365 (2015).
  * [85] G. Vallone, "Role of beam waist in Laguerre-Gauss expansion of vortex beam," Opt. Lett. 42(6), 1097–1100 (2017).
  * [86] Unlike other works, here, we also consider the radial-mode scrambling caused by atmospheric turbulence. Since the information is encoded by the photonic OAM state, we restrict our attention to the crosstalk probability within the OAM subspace.
  * [87] G. J. Fan-Yuan, C. Wang, S. Wang, Z. Q. Yin, H. Liu, W. Chen, D. Y. He, Z. F. Han, and G. C. Guo, "Afterpulse analysis for quantum key distribution," Phys. Rev. Applied 10(6), 064032 (2018).
  * [88] A. Ferenczi and N. Lütkenhaus, "Symmetries in quantum key distribution and the connection between optimal attacks and optimal cloning," Phys. Rev. A 85(5), 052310 (2012).
  * [89] J. M. Martin and S. M. Flatté, "Intensity images and statistics from numerical simulation of wave propagation in 3-D random media," Appl. Opt. 27(11), 2111–2126 (1988).
  * [90] G. Sedmak, "Implementation of fast-Fourier-transform-based simulations of extra-large atmospheric phase and scintillation screens," Appl. Opt. 43(23), 4527–4538 (2004).
  * [91] L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd edn. (SPIE, 2005).
  * [92] It is noted that the validity of ILPS is guaranteed by calculating its phase structure function. As discussed in Ref. [63], a great coincidence between the average structure function and theoretical one is commonly obtained by setting $N_{col}\geq 2$.
  * [93] A. Reeves, "Soapy: an adaptive optics simulation written purely in Python for rapid concept development," Proc. SPIE 9909, 99097F (2016).
  * [94] It is worth noting that the effect of atmospheric turbulence on beam propagation is wavelength dependent. We assumed the wavelength of beacon light is close enough to signal modes in this paper. Concretely, a beacon light whose wavelength around $100nm$ shorter than signal modes is acceptable[54, 56, 58].
  * [95] G. J. Gbur, Singular Optics, 1st edn. (CRC Press, 2017).
  * [96] D. J. Sanchez and D. W. Oesch, "Localization of angular momentum in optical waves propagating through turbulence," Opt. Express 19(25), 25388–25396 (2011).
  * [97] D. J. Sanchez and D. W. Oesch, "Orbital angular momentum in optical waves propagating through distributed turbulence," Opt. Express 19(24), 24596–24608 (2011).
  * [98] T. M. Venema and J. D. Schmidt, "Optical phase unwrapping in the presence of branch points," Opt. Express 16(10), 6985–6998 (2008).
  * [99] M. A. Herraez, D. R. Burton, M. J. Lalor, and M. A. Gdeisat, "Fast two-dimensional phase-unwrapping algorithm based on sorting by reliability following a noncontinuous path," Appl. Opt. 41(35), 7437–7444 (2002).
  * [100] J. Zhang, X. Tian, J. Shao, H. Luo, and R. Liang, "Phase unwrapping in optical metrology via denoised and convolutional segmentation networks," Opt. Express 27(10), 14903–14912 (2019).
  * [101] The realistic AO correction mentioned in this paper is the results after eliminating the wrapped cuts.
  * [102] For the weak and strong scintillation regime, we set $C_{n}^{2}=2.2\times 10^{-15}m^{-2/3}$ and $3.22\times 10^{-14}m^{-2/3}$, which is equivalent to $r_{0}=0.05m$ and $r_{0}=0.01m$ when $z=1000m$. It is noted that $r_{0}$ is calculated with a fixed wavelength of $500nm$, which is not same as the intrinsic wavelength of the incident states. The reason why we chose $500nm$ for the calculation is that $r_{0}$ is associated with wavelength according to the Eq. (3) in Ref. [51]. In atmospheric optics, a wavelength of $500nm$ is generally specified to facilitate the use of $r_{0}$ to measure the turbulence strength of a communication link.
  * [103] B. Ndagano and A. Forbes, "Characterization and mitigation of information loss in a six-state quantum-key-distribution protocol with spatial modes of light through turbulence," Phys. Rev. A 98(6), 062330 (2018).
  * [104] J. A. Anguita, M. A. Neifeld, and B. V. Vasic, "Turbulence-induced channel crosstalk in an orbital angular momentum-multiplexed free-space optical link," Appl. Opt. 47(13), 2414–2429 (2008).
  * [105] It is highlighted that, in a realistic AO system, the correction order is closely associated with the number of actuator onto a deformable mirror. We set the correction capability of AO to $30$-order for the realistic consideration.
  * [106] S. Wang, Y. Ren, R. Rao, and X. Miao, "Influence of atmosphere attenuation on quantum interferometric radar," Acta Phys. Sin. 66(15), 150301 (2017).
  * [107] The impact of aerosol-particle scattering on the performance of OAM-based QKD can be described by adjusting the size of receiving aperture. An interesting topic is that whether eavesdroppers can steal information through scattered light (see a similar discussion in Ref. [108]).
  * [108] P. Chaiwongkhot, K. B. Kuntz, Y. Zhang, A. Huang, J. P. Bourgoin, S. Sajeed, N. Lükenhaus, T. Jennewein, and V. Makarov, "Eavesdropper’s ability to attack a free-space quantum-key-distribution receiver in atmospheric turbulence," Phys. Rev. A 99(6), 062315 (2019).
  * [109] F. Wang, F. Zeng, J. Zhao, B. Braverman, Y. Zhou, M. Mirhosseini, X. Wang, H. Gao, F. Li, R. W. Boyd, and P. Zhang, "High-dimensional quantum key distribution based on mutually partially unbiased bases," Phys. Rev. A 101(3), 032340 (2020).
  * [110] Y. Yuan, D. Liu, Z. Zhou, H. Xu, J. Qu, and Y. Cai, "Optimization of the probability of orbital angular momentum for Laguerre-Gaussian beam in Kolmogorov and non-Kolmogorov turbulence" Opt. Express 26(17), 21861–21871 (2018).
  * [111] X. L. Ge, B. Y. Wang, and C. S. Guo, "Evolution of phase singularities of vortex beams propagating in atmospheric turbulence," J. Opt. Soc. Am. A 32(5), 837–842 (2015).
  * [112] G. Gbur and R. K. Tyson, "Vortex beam propagation through atmospheric turbulence and topological charge conservation," J. Opt. Soc. Am. A 25(1), 225–230 (2008).
  * [113] D. P. Greenwood, "Bandwidth specification for adaptive optics systems," J. Opt. Soc. Am. A 67(3), 390–393 (1977).
  * [114] It is noted that we generally set the bandwidth of AO as $2$–$3$ times the sample period of a Hartmann camera. Here we set to $5$ because we consider the time delay caused by phase unwrapping.
  * [115] Refs. [38, 58] indicate that a comprehensive correction can be obtained if the Greenwood time of turbulence is ten times larger than the time delay of AO.