# Physical meaning of the deviation scale under arbitrary turbulence strengths
of optical orbital angular momentum

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

###### Abstract

The recently so-called deviation scale [C. M. Mabena et al., Phys. Rev. A 99,
013828 (2019)] bridges the connection between the result of the infinitesimal
propagation equation (IPE) prediction and that of the single phase screen
(SPS) approximation. Thanks to the multiple phase screen (MPS) approach, in
this paper we elaborate the physical meaning of the deviation scale: the
spatial accumulation of slight intensity modulation of incident orbital
angular momentum (OAM) carrying beam splits the original vortex into multiple
individual vortices with a topological charge (TC) of $+1$ and re-generates
the vortex-antivortex pairs with a TC of $+1$ and with a TC of $-1$, leading
to a significant deviation between these two different results only when the
disruption of this compound effect on the phase distribution of the incident
OAM-carrying beam becomes more significant. Other than that, we also show that
the appearance of the deviation scale cannot be predicted only by the Rytov
variance, which can be predicted through the vortex-splitting ratio of the
received optical field alone or with the help of the normalized propagation
distance.

###### doi:

http://dx.doi.org/10.1364/JOSAA.XX.XXXXXX

††journal: josaa

##  1 INTRODUCTION

The property that photon carries orbital angular momentum (OAM) was firstly
proven by Allen in 1992[1], which offers a feasible way to encode information
in an infinitely large Hilbert space. It had demonstrated an impressive
significance in many applications such as modal diversity[2], optical
trapping[3, 4] and imaging[5, 6], remote sensing[7], and quantum
information[8, 9, 10]. Unfortunately, such advantages significantly degrade in
atmospheric propagation, partly because the helical phase $e^{il\phi}$ of an
OAM-carrying beam, with $l$ the quantum number indicating the amount of OAM
$l\hbar$ carried by each photon in a beam, becomes extremely fragile while
encountering atmospheric turbulence.

Recently, most previous studies, including theories[11, 12, 13, 14, 15] and
experiments[16, 17, 18, 19], have been devoted to investigate the effects of
atmospheric turbulence on an OAM-carrying beam using phase perturbation
approximation, i.e., single phase screen (SPS) approximation (we call this as
the SPS approximation in general referring to the Paterson C model proposed in
Ref. [11]). This model indicates that the measured OAM expectation value in
the output plane is the same as that of the input mode. Until 2017 Lavery et
al. found that this result does not hold for long free-space link lengths or
more turbulent links in terms of the experiment of transmitting high-
dimensional structured optical fields in an urban environment[20, 21]. More
importantly, in 2019 Mabena et al. gives a precise prediction for describing
the crosstalk evolution of atmospheric propagation under arbitrary
scintillation conditions[22] using the infinitesimal propagation equation
(IPE) of Roux[23, 24, 25, 26].

However, although a vast amount of attentions have been paid to study the
crosstalk evolution of an OAM-carrying beam propagating through a turbulent
channel over past decades, no contributions have yet elaborated the meaning of
the critical point, known as the so-called deviation scale[22], which bridges
the connection between the result of the SPS approximation and that of the IPE
prediction. It is the topic of this paper. In order to compare with Roux’s
previous work, in this paper we first employ the multiple phase screen (MPS)
approach to re-examine the correctness of the minimal set of parameters, in
other words, whether these parameters remain valid to completely determine the
crosstalk evolution considering the radial-mode scrambling. Based on this re-
examination, we believe that the results of numerical simulation are
equivalent to that of the IPE prediction. Secondly, we elucidate the physical
meaning of the deviation scale through calculating the vortex-splitting ratio
of the received optical field.

The SPS approximation typically follows the assumption that scintillation is
weak enough to neglect the diffraction effect in the atmospheric
propagation[27]. This approximation can be considered as the result of
geometrical optics. Ironically, we found that the physical meaning of the
deviation scale can be elaborated clearly when the assumption is removed and
intensity fluctuations are considered, which can be accomplished by using so-
called split-step beam propagation method[28] that is valid in all
scintillation conditions. Besides, we also reveal that the deviation begins to
appear only when the vortex-splitting ratio is beyond a specific threshold.
The interrogation of the nature of the deviation scale may be advantageous for
us to determine how and when we can employ a suitable approximation or method
to investigate the effects of atmospheric turbulence on an OAM-carrying beam.

A qualitative explanation can be achieved from the difference curves between
the results of the IPE prediction and that of the SPS approximation. When the
Rytov variance is less than unity and the normalized propagation distance is
beneath a specific threshold, no difference is found between these two
results. Conversely, the deviation starts to appear when the distance exceeds
this threshold. From the quantitative prospective, the spatial accumulation of
the intensity modulation of an incident OAM-carrying beam becomes more severe
with the increasing propagation distance, which splits the original vortex
into different individual vortices with a topological charge (TC) of $+1$[29]
and re-generates vortex-antivortex pairs with a TC of $+1$ and a TC of
$-1$[34, 35, 36], leading to the appearance of deviation scale only when the
disruption of this compound effect on the phase distribution of the incident
OAM-carrying beam becomes more significant. In fact, more precise thresholds
that determine the appearance of the deviation scale for different azimuthal
indices of an OAM-carrying beam are obtained through quantitative analysis.
Other than that, we conclude that the appearance of deviation scale cannot be
predicted only by the Rytov variance, which can be predicted through the
vortex-splitting ratio of the received optical field alone or with the help of
the normalized propagation distance.

The overall structure of the main text is organized as follows. In Sec. 2, we
briefly introduce the theoretical model of the crosstalk evolution of an OAM-
carrying beam using so-called Laguerre-Gaussian (LG) modes. In Sec. 3, we
first review several different physical quantities defined in Roux’s previous
work[22, 37, 38, 39, 40]; then provide some technical aspects of our numerical
simulation; finally reexamine whether the minimal set of parameters obtained
by the IPE remains valid to completely determine the crosstalk evolution
considering the radial-mode scrambling. Based on the results given in above
section, Sec. 4 presents a qualitative and a quantitative explanation for the
nature of the deviation scale. Sec. 5 we conclude.

Figure 1: The source generates a single LG mode and sends it through a
turbulent atmosphere (modeled by multiple turbulent cells 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 $w_{0}$ of the LG
modes constant[41].

##  2 DESCRIPTION OF THE CROSSTALK EVOLUTION

The conceptual diagram of optical system that is simulated by our numerical
procedure is shown in Fig. 1. Without loss of generality, we suppose that $z$
is the propagation distance, and the incident OAM-carrying beam is a single LG
mode with azimuthal index $l_{0}$ and radial index $p_{0}$, which can be
expressed in cylindrical coordinates by

$\displaystyle LG_{p_{0},l_{0}}^{\left(q_{0}\right)}\left(r,\phi,0\right)$
$\displaystyle=A\left(\frac{i\sqrt{kz_{R}}r}{q_{0}}\right)^{\left|l_{0}\right|+1}L_{p_{0}}^{\left|l_{0}\right|}\left(\frac{kz_{R}r^{2}}{\left|q_{0}\right|^{2}}\right)$
$\displaystyle\left(-\frac{q_{0}^{\ast}}{q_{0}}\right)^{p_{0}}\frac{e^{-\frac{ikr^{2}}{2q_{0}}}}{r}e^{il_{0}\phi},$
(1)

with $L_{p_{0}}^{\left|l_{0}\right|}\left(\cdot\right)$ as the generalized
Laguerre polynomial, where
$A=\sqrt{p_{0}!/\pi\left(\left|l_{0}\right|+p_{0}\right)!}$ is the
normalization constant and $q_{0}=iz_{R}$ represents the complex parameter
associated with beam waist. $z_{R}=\pi w_{0}^{2}/\lambda$ and $k=2\pi/\lambda$
denote the Rayleigh range and the wave number respectively, $\lambda$ is the
wavelength and $w_{0}$ is the beam waist at input plane.

Beam propagating through a turbulent channel is often described by the split-
step beam propagation method[28]. It generally means that, as shown in Fig. 1,
a turbulent channel can be divided into a series of turbulent cells, each cell
introduces a random contribution $\phi$ to the phase, but essentially no
change in the amplitude; besides, intensity fluctuations build up by
diffraction over many cells. To agree well with the phase structure function
of the Kolmogorov turbulence model, random phase screen lost low spatial
frequencies need to be compensated using subharmonic method[42].

With this in mind, the split-step method is used to simulate the procedure of
atmospheric propagation instead of the SPS approximation. Since LG modes with
same $w_{0}$ form an orthonormal basis, the complex quantity $q_{0}$ is
modified to $q\left(z\right)=z+q_{0}$ during the spectral decomposition for
attributing the intermodal crosstalk to the impact of turbulence entirely,
where the notation $q\left(z\right)$ called the Siegman complex parameter[43].
Hence, the received optical field can be decomposed into a superposition of
several LG modes

$U_{p,l}\left(r,\phi,z\right)=\sum_{p=0}^{\infty}\sum_{l=-\infty}^{\infty}c_{p,l}LG_{p,l}^{\left(q\right)}\left(r,\phi,z\right),$
(2)

with the coefficient $c_{p,l}$ calculating from the overlap integral

$c_{p,l}=\int_{0}^{\infty}rdr\int_{0}^{2\pi}d\phi
U_{p,l}\left(r,\phi,z\right)LG_{p,l}^{\left(q\right)\ast}\left(r,\phi,z\right),$
(3)

where the asterisk denotes the complex conjugate. Besides, the fraction of
optical power that is transferred from the input mode with azimuthal index
$l_{0}$ to other modes with azimuthal index $l$ can be described as

$P\left(l\right)=\sum_{p=0}^{\infty}\left|c_{p,l}\right|^{2},$ (4)

From the above analysis, we summary the effects of atmospheric turbulence on a
LG mode with more intuitive way: the optical power of the input mode with
azimuthal index $l_{0}$ is carved into unlimited pieces in terms of the power
fraction $\left|c_{p,l}\right|^{2}$; each piece of LG modes attachs a
turbulence-induced random phase in terms of the argument of the overlap
intergral, namely, $\arg\left(c_{p,l}\right)$.

##  3 NUMERICAL REEXAMINATION OF THE IPE

###  3.1 Several parameters

To reexamine whether the minimal set of parameters obtained by the IPE remains
valid to completely determine the crosstalk evolution considering the radial-
mode scrambling, we briefly review several dimensionless quantities that are
theoretically indispensable in the result of the IPE prediction: the
normalized propagation distance $t\equiv z/z_{R}$, the normalized turbulence
strength $K\equiv C_{n}^{2}w_{0}^{11/3}\pi^{3}/\lambda^{3}$ and the relative
beam width $W\equiv w_{0}/r_{0}$, where $C_{n}^{2}$ is the refractive index
structure constant, $r_{0}$ is the Fried parameter. Besides, the compound
quantities $W$, $K$, and $t$ are related by[22, 38, 39]

$W=1.37\left(Kt\right)^{3/5},$ (5)

The intensity fluctuation of an optical field propagating through atmospheric
turbulence is often centered around the scintillation index (For a more
general expression of scintillation index, we refer the reader to Ref. [44]),
which is the normalized variance of intensity fluctuation, and can be defined
as[27]

$\sigma_{I}^{2}\equiv\frac{\left\langle I^{2}\right\rangle-\left\langle
I\right\rangle^{2}}{\left\langle I\right\rangle^{2}}=\frac{\left\langle
I^{2}\right\rangle}{\left\langle I\right\rangle^{2}}-1,$ (6)

where the notation $I$ and $\left\langle\cdot\right\rangle$ represent the
intensity of an optical field and ensemble average, respectively. Under the
weak scintillation condition based on the Kolmogorov turbulence model, the
scintillation index of a plane wave can be expressed by[27, 45]

$\sigma_{R}^{2}=1.23C_{n}^{2}k^{7/6}z^{11/6},$ (7)

which can also be used to describe the intensity fluctuation over the
turbulent link when extended to strong scintillation condition by increasing
either $C_{n}^{2}$ or $z$, or both. The link between these two quantities is
that under the weak scintillation condition, the scintillation index is
proportional to the Rytov variance for a plane wave. In other words, the
scintillation index increases with increasing values of the Rytov variance
until it reaches a maximum value greater than unity in the regime
characterized by random focusing[45]. Hence, we employ the Rytov variance as
an indicator of the scintillation strength to avoid the circumstance that one
scintillation index corresponds to two scintillation conditions. The Rytov
variance can be also expressed in terms of $K$ and $t$[22, 39]

$\sigma_{R}^{2}=1.64W^{5/3}t^{5/6}=1.055W^{55/18}/K^{5/6},$ (8)

###  3.2 Data details and notes

The result of numerical simulation in one iteration, representing a single
realization of turbulence, can be repeated several times. The ensemble average
of different iterations of the two steps represents the result of crosstalk
evolution under one fixed compound quantity (i.e., $K$ or $t$). Table 1 gives
the different values of $K$ used in our simulation. For convenience of
comparison, the turbulence strength $C_{n}^{2}$, the beam waist $w_{0}$, and
the wavelength $\lambda$ are specifically assigned to guarantee that $W$ is
limited to a certain range. Therefore, sampling intervals and propagation
distances vary with the variation of $K$ in Table 1 during these simulations.

Eq. (8) gives the relationship between $\sigma_{R}^{2}$ and $K$. Hence, it can
be observed that different scintillation conditions can be acquired during the
numerical simulations by adjusting this compound quantity with fixed $W$.
Further, since the spacing between two turbulent cells varies smartly with
these parameters, different numbers of LG modes used for spectral
decomposition are set in these simulations. Finally, it is worth highlighting
that the total propagation distance ranges from a few meters to hundred
kilometers.

###  3.3 Evolution rule under arbitrary scintillation conditions

The fraction of optical power obtained in the output plane that remains in the
input LG mode (we call this as the eigen-crosstalk probability in this paper;
(a1) $l_{0}=0$, (b1) $l_{0}=1$, (c1) $l_{0}=2$, (d1) $l_{0}=3$) is illustrated
in Fig. 2. All plots are plotted as a function of $W$. Solid curve in all
plots correspond to the results of the SPS model using quadratic
approximation[46], while the results of our numerical results are shown with
different shaped lines, corresponding to different values of $K$ given in
Table 1. According to Eq. (8), the Rytov variances in $K=0.05$ are shown in
all right-column plots with dashed lines. The dotted lines with small-spacing
represent the deviation scale, while the large-spacing ones represent the
scale determining the onset of strong scintillation. The horizontally dashed-
dotted lines correspond to $\sigma_{R}^{2}=1$.

Table 1: Different values of the compound quantity $K$ composed of beam parameters and turbulence parameters shown in Fig. 2. $K$ | $C_{n}^{2}\left(m^{-2/3}\right)$ | $w_{0}\left(m\right)$ | $\lambda\left(nm\right)$
---|---|---|---
$0.05$ | $10^{-16}$ | $0.05$ | $1000$
$0.1$ | $10^{-16}$ | $0.07$ | $1214$
$0.3$ | $10^{-16}$ | $0.1$ | $1314$
$0.8$ | $5\times 10^{-16}$ | $0.05$ | $689$
$2$ | $5\times 10^{-16}$ | $0.08$ | $900$
$5$ | $5\times 10^{-15}$ | $0.05$ | $807$
$20$ | $10^{-14}$ | $0.05$ | $640$
$100$ | $10^{-13}$ | $0.08$ | $1432$
$1000$ | $10^{-12}$ | $0.08$ | $1434$
$10^{4}$ | $5\times 10^{-12}$ | $0.1$ | $1495$
Figure 2: The eigen-crosstalk probabilities averaged over 300 realizations of
turbulence as a function of $W$ under different values of normalized
turbulence strengths $K$ given in Table 1 for (a1) $l_{0}=0$, (b1) $l_{0}=1$,
(c1) $l_{0}=2$, (d1) $l_{0}=3$. The solid lines represent the results of the
SPS approximation. The Rytov variances in $K=0.05$ (weak turbulence) are
provided in the right-column plots. The short-spacing dotted lines and long-
spacing dotted lines represent the deviation scale and the scale determining
the onset of strong scintillation, respectively. For each azimuthal index
$l_{0}$, the deviation scale appears at a smaller value compared to the scale
determining the onset of strong scintillation. The horizontally dashed-dotted
lines in all right-column plots correspond to $\sigma_{R}^{2}=1$. The solid
line and the square scatter points in the right-column plots ((a2) $l_{0}=0$;
(b2) $l_{0}=1$; (c2) $l_{0}=2$; (d2) $l_{0}=3$) is the partial enlargement of
the results presented in the left-column plots.

We observe that for large values of $K$ (e.g., $K\gtrsim 20$ for $l_{0}=0,1$
and $K\gtrsim 100$ for $l_{0}=2,3$), the evolution curves of the eigen-
crosstalk probability obtained by our numerical simulation almost coincide
with the result of the SPS approximation. However, this coincidence does not
exist as $K$ becomes smaller (e.g., $K\lesssim 5$ for $l_{0}=0,1$ and
$K\lesssim 20$ for $l_{0}=2,3$), which means that another compound quantity
$K$ is needed to be added to describe the evolution of eigen-crosstalk
probability except $W$. Moreover, we found that the eigen-crosstalk
probability of input modes with a larger azimuthal index ($l_{0}\gtrsim 2$)
decreases faster compared to that of the smaller one, more obviously when
$K\lesssim 2$, which is likely because LG modes with a larger $l_{0}$ possess
a larger beam waist so that it’s more susceptible to atmospheric turbulence.

The deviation phenomenon between the results of numerical simulation and that
of the SPS approximation under arbitrary turbulence strengths is also shown in
Fig. 2. We zoom in the results of numerical simulation of $K=0.05$ in the
right-column of Fig. 2 ((a2) $l_{0}=0$, (b2) $l_{0}=1$, (c2) $l_{0}=2$, (d2)
$l_{0}=3$). The crossing points of the numerical curves and the large-spacing
dotted lines represent the onset of strong scintillation, while the crossing
points of the numerical curves and the short-spacing dotted lines indicate at
what values of the Rytov variance the deviation occur. Notably, the results in
Fig. 2 seem to indicate that LG modes with a larger azimuthal index have a
smaller deviation scale. Further, it reveals a well-known conclusion that the
deviation scale is always smaller than the scale determining the onset of
strong scintillation[22]. However, why are they different? more
straightforwardly, what is the nature of the deviation scale? It deserves to
be investigated, and it is also the topic of this paper.

###  3.4 Verification of the minimal set of parameters

The evolution curves of eigen-crosstalk probability as a function of $W$ for
$l_{0}=0$ are illustrated in Fig. 3. These curves are obtained in $K=0.066$
with different combinations of dimension parameters shown in Table 2. The
smaller $K$ is chosen because it represents a condition where the evolution of
eigen-crosstalk probability deviates significantly from that of the SPS
approximation[38]. We observe that although different sets of parameters are
selected in the numerical simulation, all of curves coincide under the same
$K$, which means all kinds of beam parameters and turbulence parameters (i.e.,
$C_{n}^{2}$, $z$, $w_{0}$ and $\lambda$) always appear in the form of compound
quantities (i.e., $K$, $t$, and $W$) within the crosstalk evolution calculated
by the Kolmogorov turbulence model under arbitrary scintillation conditions.
Compared to the minimal set of parameters that decides the result of the IPE
prediction, one can conclude that these parameters remain valid to completely
determine the crosstalk evolution considering the radial-mode scrambling.

Figure 3: The eigen-crosstalk probability as a function of $W$ with the same normalized turbulence strength $K=0.066$ given in Table 2 for $l_{0}=0$, averaged over 300 realizations of turbulence. Each set of dimensionless parameters are combined with different beam and turbulence parameters. The error bars represent the standard error. Table 2: Different combinations of beam parameters and turbulence parameters for $K=0.066$ shown in Fig. 3. $set$ | $C_{n}^{2}\left(m^{-2/3}\right)$ | $w_{0}\left(m\right)$ | $\lambda\left(nm\right)$
---|---|---|---
$1$ | $10^{-17}$ | $0.1$ | $1000$
$2$ | $10^{-16}$ | $0.05$ | $924$
$3$ | $5\times 10^{-18}$ | $0.1$ | $794$
$4$ | $5\times 10^{-17}$ | $0.08$ | $1302$
$5$ | $10^{-18}$ | $0.13$ | $640$
Table 3: Different values of the compound quantity $t$ composed of beam parameters and turbulence parameters shown in Fig. 4. $t$ | $z\left(km\right)$ | $w_{0}\left(m\right)$ | $\lambda\left(nm\right)$
---|---|---|---
$2\times 10^{-4}$ | $5\times 10^{-3}$ | $0.1$ | $1263$
$10^{-3}$ | $3\times 10^{-2}$ | $0.08$ | $672$
$10^{-2}$ | $0.1$ | $0.06$ | $1169$
$0.1$ | $1.2$ | $0.05$ | $658$
$0.5$ | $5$ | $0.05$ | $791$
$1$ | $22.5$ | $0.08$ | $900$
$1.5$ | $30$ | $0.07$ | $770$
$2$ | $50$ | $0.08$ | $806$
$3$ | $70$ | $0.08$ | $864$
$5$ | $100$ | $0.08$ | $1000$
$10$ | $200$ | $0.08$ | $1000$

##  4 PHYSICAL MEANING OF THE DEVIATION SCALE

###  4.1 Qualitative understanding

From above analysis, we highlight that compound quantities composed of $K$ and
$W$ can completely determine the crosstalk evolution caused by atmospheric
turbulence. Consequently, it not necessarily investigates the crosstalk
evolution under different values of normalized propagation distance $t$.
However, although $K$, $t$ and $W$ are pairwise independent compound
quantities, it may be help us to give a qualitative understanding of the
deviation scale by studying the influence of $t$ on the eigen-crosstalk
probability.

To better determine the value of $\sigma_{R}^{2}$ and $t$ where the results of
the IPE prediction (we believe the results of numerical simulation are
equivalent to that of the IPE prediction due to the validity of the minimal
set of parameters) start to deviate from that of the SPS approximation, in
Fig. 4 we illustrate how the difference of eigen-crosstalk probability between
these two results, namely, $\Delta P\equiv P_{SPS}-P_{IPE}$, changes with
respect to $\sigma_{R}^{2}$ under different values of $t$ shown in Table 3.
For convenience of analysis, Fig. 4 is divided into four areas. We see that
the points where the IPE predictions start to deviate from the SPS
counterparts occur at $\sigma_{R}^{2}\simeq 0.1$, especially for the larger
value of $t$. Further, we observe that when $0.1<\sigma_{R}^{2}<1$ and
$t<0.5$, the deviation between these two results remains still very small. On
the contrary, if the above conditions do not hold any more, the deviation will
grow rapidly with the increase of $\sigma_{R}^{2}$ even if the turbulent
channel is still under the weak scintillation condition.

We mark a series of cruciform scatter points in Fig. 4, which represents the
results obtained in $K=0.05$. Based on these results, we can observe clearly
why the deviation always appears at smaller values than the scale determining
the onset of strong scintillation. That is because the difference of eigen-
crosstalk probability in $K=0.05$ have exceeded the above threshold of $t$ in
the regime of $0.1<\sigma_{R}^{2}<1.$ Actually, this effect can be seen more
clearly when Fig. 2 is plotted as a function of $t$ instead of $W$. Hence, we
can qualitatively conclude that the SPS approximation is valid if
$\sigma_{R}^{2}<1$ and $t<0.5$ for LG modes.

Figure 4: The difference curves of eigen-crosstalk probability between the
results of the IPE prediction and that of the SPS approximation, namely,
$\Delta P\equiv P_{SPS}-P_{IPE}$, averaged over 300 realizations of turbulence
as a function of $\sigma_{R}^{2}$ under different normalized propagation
distances $t$ given in Table 3 for (a) $l_{0}=0$, (b) $l_{0}=1$, (c)
$l_{0}=2$, (d) $l_{0}=3$. Each graph is divided by four areas composed of
$\sigma_{R}^{2}<0.1$, $0.1<\sigma_{R}^{2}<1$, $1<\sigma_{R}^{2}<10$ and
$\sigma_{R}^{2}>10$. A series of cruciform scatter points are marked,
representing the results obtained in $K=0.05$.

Other than the interrogation of the nature of the deviation scale, we can also
conclude some other interesting phenomena from the results in Fig. 4, which
can be summarized as follows: When the scintillation condition is beyond the
weak range (i.e., $1<\sigma_{R}^{2}<10$), the deviation between two results
becomes smaller as the increase of $\sigma_{R}^{2}$. When $\sigma_{R}^{2}>10$,
these deviations together incline to disappear, which is likely because of the
effect of scintillation saturation.

###  4.2 Quantitative understanding

What can be seen from the above analysis is that the deviation between the
results of the IPE prediction and that of the SPS approximation comes from the
spatial accumulation of the intensity modulation of the input LG mode. In
other words, for a LG mode that are perturbed while propagating over a series
of turbulent cells the compound effect on the LG mode is more damaging than
the effect of simply accumulating the phase perturbation over the link. The
direct result of the intensity modulation changes the location of the original
vortex. Hence, we anticipate that even if the turbulent channel is under the
weak scintillation condition, the spatial accumulation of slight intensity
modulation of the input LG mode splits the vortex into multiple individual
vortices with a TC of $+1$ and re-generates the vortex-antivortex pairs with a
TC of $+1$ and with a TC of $-1$, which may be the main reason for the
appearance of the deviation scale.

To examine our anticipation, Fig. 5 illustrates the evolution curves of the
eigen-crosstalk probability, the average OAM and vortex-splitting ratio[20,
21] in the output plane, all of which are calculated under $K=0.05$ and
different values of $l_{0}$. Besides, all plots are plotted as a function of
$W$. Notably, the reason why we choose the vortex-splitting ratio as an
indicator to examine our anticipation is that for LG modes with a specific
azimuthal index, this parameter contains much information about the vortex
distribution in the phase of the received optical field, which characterizes
all the effects of atmospheric propagation under the weak scintillation
condition. Considering atmospheric turbulence can also generate vortex-
antivortex pairs, the vortex-splitting ratio can be re-evaluated by[36]

$\nu\equiv\frac{1}{Nw_{0}}{\displaystyle\sum\limits_{i=1}^{N}}q_{i}V_{r}^{\left(i\right)}.$
(9)

where $N$ represents the total number of vortices in the output plane. Each
vortex is identified by the index $i$. $q_{i}$ and $V_{r}^{\left(i\right)}$
stand for the TC of the vortex with index $i$ and the radial distance from the
beam origin for the vortex with index $i$, respectively. We have to emphasize
that this definition not only takes account into the turbulence-induced
vortex-antivortex pairs but also includes the effect of vortex-splitting. Our
vortex detection algorithm is a modified version of the algorithm proposed in
Refs. [47, 48, 49, 50].

Figure 5: The eigen-crosstalk probabilities ((a)–(c)), the vortex-splitting
ratio and the average OAM ((d)–(f)), averaged over 300 realizations of
turbulence as a function of $W$ with $K=0.05$: (a), (d) $l_{0}=1$, (b), (e)
$l_{0}=2$, (c), (f) $l_{0}=3$. The standard errors are plotted by the shaded
area around each curve. The solid lines represent the results of the SPS
approximation. The Rytov variance in $K=0.05$ are depicted in dashed line.
(a)–(c) are horizontally divided into two areas composed of $\sigma_{R}^{2}<1$
and $\sigma_{R}^{2}>1$ using dashed-dotted lines. The splitting-thresholds
that represent the appearance of the deviation scale for different azimuthal
indices of LG modes are plotted by the dotted line in (d)–(f): (d) $\nu=0.1$
for $l_{0}=1$, (e) $\nu=0.15$ for $l_{0}=2$, (f) $\nu=0.2$ for $l_{0}=3$.
These splitting-thresholds can be easily translated into the threshold of $t$
with different values: (d) $t=0.648$ for $l_{0}=1$, (e) $t=0.369$ for
$l_{0}=2$, (f) $t=0.255$ for $l_{0}=3$.

We observe in Fig. 5 that for different azimuthal indices of LG modes and
specific turbulence strength, the phenomena of vortex splitting and vortex-
antivortex pairs re-generation occur in the output plane at the early stages
of propagation (e.g., for $l_{0}=1$, atmospheric turbulence steers the
original vortex off beam axis and leads to the vortex-antivortex pairs re-
generation). Other than that, we see that the compound effect caused by
atmospheric turbulence, quantified by the vortex-splitting ratio, becomes more
pronounced progressively as $W$ increases. This behavior can be seen more
clearly when the curves are plotted as a function of $t$ instead of $W$. Based
on the results presented in Fig. 5, we found that with the increase of the
vortex splitting ratio (e.g., $\nu\geq 0.1$ for $l_{0}=1$, $\nu\geq 0.15$ for
$l_{0}=2$ and $\nu\geq 0.2$ for $l_{0}=3$), the average OAM and the eigen-
crosstalk probabilities gradually deviate from the results of the SPS
approximation. A suitable explanation is that when the turbulent channel is
under the weak scintillation condition, the turbulence-induced intensity
fluctuation splits the original vortex into multiple vortices with a TC of
$+1$ in the output plane, and possibly accompanied by the re-generation of
vortex-antivortex pairs that carry opposite-sign unity TC. After undergoing
the spatial accumulation of a considerable amount of compound effect, the
probability of detecting the TC of $+1$ in the phase distribution of the
received optical field becomes more larger, which leads to a significant
reduction of the average OAM and the deviation between the results of the IPE
prediction and that of the SPS approximation.

Therefore, we believe the appearance of the deviation is actually the process
of the spatial accumulation of the compound effect. From the results presented
in Fig. 5 (see the rectangular shaded areas in all six plots), it is worth
highlighting that such phenomena will happen only if the vortex-splitting
ratio of the received optical field is beyond a specific threshold (we call
this as the splitting-threshold, e.g., $\nu=0.1$ for $l_{0}=1$, $\nu=0.15$ for
$l_{0}=2$ and $\nu=0.2$ for $l_{0}=3$). Other than that, it is not difficult
to find that different splitting-thresholds that we obtained in fact provide a
more precise application scope of the SPS approximation for LG modes with a
specific azimuthal index. To this end, we calculate the values of normalized
propagation distance $t$ that correspond to the aforementioned splitting-
thresholds by Eq. (5), namely, $t=0.648$ for $l_{0}=1$, $t=0.369$ for
$l_{0}=2$ and $t=0.255$ for $l_{0}=3$. The reason why $t$ becomes smaller as
$l_{0}$ increases is that LG modes with a larger azimuthal index possess a
larger beam waist so that it’s more susceptible to atmospheric turbulence.

On the other hand, we also verify our anticipation in Fig. 6 with a more
straightforward way, which is a evident illustration of the phase
distributions of the received optical field. The OAM spectra of the received
optical field, averaged over 300 realizations of turbulence under the weak
scintillation condition, are also plotted in Fig. 6. The top row in Fig. 6
represents the results below the splitting-threshold (e.g., (a) $\nu=0.0337$,
(b) $\nu=0.1112$, (c) $\nu=0.1968$, corresponding to $t=0.157$ approximately),
while the bottom row represents the results above the splitting-threshold
(e.g., (d) $\nu=0.2603$, (e) $\nu=0.4447$, (f) $\nu=0.599$, corresponding to
$t=1.376$ approximately). By comparing the results between top and bottom, we
also obtain the same conclusions presented in Fig. 5, which is exactly the
kind of result we expected. More importantly, we observe in the top row of
Fig. 6 that a pair of vortex-antivortex pairs occurs in the phase distribution
of the received optical field even when the vortex-splitting ratio is below
the splitting-threshold (see Fig. 6(c)), which makes us more confident to the
above physical interpretation of the deviation scale.

At last, we conclude that the appearance of deviation scale cannot be
predicted only by the Rytov variance, which can be predicted through the
vortex-splitting ratio of the received optical field alone or with the help of
$t$. In other words, it is not completely scientific for us to employ the SPS
approximation to simulate the crosstalk evolution of atmospheric propagation
even when the turbulent channel is under the weak scintillation condition.

Figure 6: OAM spectra averaged over 300 realizations of turbulence and phase
distributions of the received optical field for a single realization of
turbulence with different azimuthal indices: (a), (d) $l_{0}=1$, (b), (e)
$l_{0}=2$, (c), (f) $l_{0}=3$. The top row represents the results below the
splitting-thresholds ((a) $\nu=0.0337$; (b) $\nu=0.1112$; (c) $\nu=0.1968$),
corresponding to $t=0.157$ approximately. The bottom row represents the
results above the splitting-thresholds ((d) $\nu=0.2603$; (e) $\nu=0.4447$;
(f) $\nu=0.599$), corresponding to $t=1.376$ approximately. The colorbar for
all phase distributions (a)–(f) are the same. The dashed lines denote the
eigen-crosstalk probabilities obtained from the results of the SPS
approximation and the arrows point to the eigen-crosstalk probabilities
obtained from the results of the IPE prediction. The bi-color dots (i.e.,
black dots and white dots) plotted in each partial enlarged plot indicate the
location of individual vortices, corresponding to a TC of $+1$ and a TC of
$-1$, respectively.

##  5 CONCLUSION

In this paper, we give a suitable physical interpretation for the recently so-
called deviation scale [C. M. Mabena et al., Phys. Rev. A 99, 013828 (2019)],
which bridges the connection between the result of the IPE prediction and that
of the SPS approximation. Before we explain the nature of the deviation scale,
we reexamine whether the minimal set of parameters obtained by the IPE remains
valid to completely determine the crosstalk evolution considering the radial-
mode scrambling. The process of re-examination extends the results of the IPE
and proofs the validity of our numerical simulation. In our endeavor to
present a qualitative understanding of the deviation scale, we found that when
the scintillation condition, measured by the Rytov variance, is weak, namely,
$\sigma_{R}^{2}<1$ and the normalized propagation distance is below than a
specific threshold, namely, $t<0.5$, the evolution curve of eigen-crosstalk
probability almost coincides with that of the SPS approximation. However, when
$t>0.5$, the deviation scale starts to appear and becomes larger as
$\sigma_{R}^{2}$ increases, which qualitatively gives a rough application
scope of the SPS approximation for an OAM-carrying beam propagating through
atmospheric turbulence. Furthermore, we demonstrate what the deviation scale
actually represents from some quantitative calculation. Our quantitative
understanding is that the spatial accumulation of slight intensity modulation
of the incident OAM-carrying beam splits the original vortex into multiple
individual vortices with a TC of $+1$ and re-generates the vortex-antivortex
pairs with a TC of $+1$ and with a TC of $-1$, leading to a significant
reduction of the value of the average OAM in the output plane and the
deviation between the results of the IPE prediction and that of the SPS
approximation. Notably, such phenomena will happen only if the disruption of
this compound effect on the phase distribution of the incident OAM-carrying
beam becomes more significant, which can be quantified by vortex-splitting
ratio of the received optical field. Or to put in another way, such phenomena
will happen only if the vortex-splitting ratio is beyond the splitting-
threshold (e.g., $\nu=0.1$ for $l_{0}=1$, $\nu=0.15$ for $l_{0}=2$ and
$\nu=0.2$ for $l_{0}=3$). In fact, the aforementioned two interpretations are
equivalent because the qualitative description gives a complete parameter
condition (i.e., $\sigma_{R}^{2}$ and $t$) composed of beam parameters and
turbulence parameters (i.e., $C_{n}^{2}$, $z$, $w_{0}$ and $\lambda$) to
determine the appearance of the deviation scale, while the quantitative one is
obtained from the splitting-threshold condition that completely and
comprehensively evaluates the results of the compound effect in the output
plane, which can be easily translated into the threshold of $t$ with different
values (e.g., $t=0.648$ for $l_{0}=1$; $t=0.369$ for $l_{0}=2$; $t=0.255$ for
$l_{0}=3$). Generally, the quantitative calculation offers a more precise
application scope of the SPS approximation. On the other hand, these
conclusions also reveal that the appearance of deviation scale cannot be
predicted only by the Rytov variance, which can be measured with the help of
$t$ or predicted through the vortex-splitting ratio of the received optical
field alone.

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).

Acknowledgment. We would like to thank the anonymous reviewers for their
valuable comments, which significantly improves the presentation of this
paper. We acknowledge the helpful discussion with Professor Filippus Stef
Roux. Zhiwei Tao thanks Professor Ruizhong Rao for his insightful suggestions.

Disclosures. The authors declare no conflicts of interest.

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