# Vector mode decay in atmospheric turbulence: a quantum inspired analysis

Isaac Nape  School of Physics, University of the Witwatersrand, Private Bag 3,
Johannesburg 2050, South Africa  Nikiwe Mashaba  School of Physics, University
of the Witwatersrand, Private Bag 3, Johannesburg 2050, South Africa  Optronic
Sensor Systems, Defence and Security, Council for Scientific and Industrial
Research (CSIR), Pretoria 0001, South Africa  Nokwazi Mphuthi  School of
Physics, University of the Witwatersrand, Private Bag 3, Johannesburg 2050,
South Africa  Sruthy Jayakumar  Department of Electrical Engineering, Indian
Institute of Technology Madras, Chennai – 600036, India  Shanti Bhattacharya
Department of Electrical Engineering, Indian Institute of Technology Madras,
Chennai – 600036, India  Andrew Forbes<EMAIL_ADDRESS>School of
Physics, University of the Witwatersrand, Private Bag 3, Johannesburg 2050,
South Africa

###### Abstract

Vector beams are inhomogeneously polarized optical fields with nonseparable,
quantum-like correlations between their polarisation and spatial components,
and hold tremendous promise for classical and quantum communication across
various channels, e.g. the atmosphere, underwater, and in optical fibre. Here
we show that by exploiting their quantum-like features by virtue of the non-
separability of the field, the decay of both the polarisation and spatial
components can be studied in tandem. In particular, we invoke the principle of
channel state duality to show that the degree of nonseparability of any vector
mode is purely determined by that of a maximally nonseparable one, which we
confirm using orbital angular momentum (OAM) as an example for topological
charges of $\ell=1$ and $\ell=10$ in a turbulent atmosphere. A consequence is
that the well-known cylindrical vector vortex beams are sufficient to predict
the behaviour of all vector OAM states through the channel, and find that the
rate of decay in vector quality decreases with increasing OAM value, even
though the spread in OAM is opposite, increasing with OAM. Our approach offers
a fast and easy probe of noisy channels, while at the same time revealing the
power of quantum tools applied to classical light.

##  I Introduction

Structured light has become topical of late Rubinsztein-Dunlop et al. (2016),
with so called cylindrical vector vortex (CVV) beams Zhan (2009) taking centre
stage in numerous fundamental and applied studies Ndagano et al. (2017a);
Rosales-Guzmán et al. (2018). For example, they form a family of natural
solutions of free-space and optical waveguides, and have been used in optical
trapping Li et al. (2019); Zhang et al. (2010); Bhebhe et al. (2018),
metrology Berg-Johansen et al. (2015), as well as high capacity classical
Milione et al. (2015); Zhao and Wang (2015); Willner (2018) and quantum de
Oliveira et al. (2020); Sit et al. (2017); Cozzolino et al. (2019); Parigi et
al. (2015) communication. To meet the demand of such growing applications, a
plethora of generation methods have emerged, including directly from lasers
Sun et al. (2012); Naidoo et al. (2016); Sroor et al. (2020), or externally
with liquid crystal q-plate technology Marrucci (2013), metasurfaces Devlin et
al. (2017); Mueller et al. (2017), and spatial light modulators Rosales-Guzmán
et al. (2020). Detection has likewise matured to include deterministic
detectors incorporating interferometers Nagali et al. (2010); Slussarenko et
al. (2010), mode sorters Walsh (2016) or both Ndagano et al. (2017b), as well
as fast digital Stokes measurements Selyem et al. (2019); Manthalkar et al.
(2020); Singh et al. (2020) and direct measures of the nonseparability or
vector quality factor McLaren et al. (2015); Ndagano et al. (2016), giving a
quantitative measure of how vector the vector beam is.

An open challenge in the context of classical and quantum communication is the
propagation of such modes through media exhibiting spatially dependent
perturbations. These might include thermal effects due to overheating of
optical elements Whinnery et al. (1967), rapid refractive index fluctuations
in the atmosphere Andrews and Phillips (2005); Cheng et al. (2009) and
underwater Hufnagel et al. (2020), and in optical fibre Ndagano et al. (2015);
Chen and Wang (2017). In particular, atmospheric turbulence leaves the
polarisation of optical beams undisturbed while the spatial components degrade
rapidly resulting in modal scattering and therefore information loss Roux
(2011); Ibrahim et al. (2013); Tyler and Boyd (2009); Smith and Raymer (2006);
Leonhard et al. (2018); Pors et al. (2011); Malik et al. (2012); Cox et al.
(2016). Consequently, information encoding with the spatial components of
light is restricted to only several km Krenn et al. (2015, 2014). In the case,
of vector beams, having coupled polarisation and spatial components, the
polarisation fields are indirectly impacted Chen and Yang (2016), resulting in
the decay of nonseparability or vectorness Ndagano et al. (2017c).

Here we exploit parallels between nonseparability of vector beams and
entanglement in quantum systems Ndagano et al. (2017c); Aiello et al. (2015);
Töppel et al. (2014); Forbes et al. (2019); Spreeuw (1998); Eberly et al.
(2016); Konrad and Forbes (2019); Toninelli et al. (2019) to deploy a quantum
toolkit for the study of vector beams in atmospheric turbulence. Importantly,
we recognise that just as the degree of entanglement of any pure quantum state
can be determined by that of a maximally entangled state Konrad et al. (2008),
courtesy of the Choi-–Jamiołkowski isomorphism (channel state duality)
Jamiołkowski (1972); Choi (1975), so it must be true that the dynamics of any
vector beam in a one-sided noisy channel should be able to be inferred from
the dynamics of just one beam, a purely inhomogeneously polarized field (a
perfectly “vector” beam with orthogonal spatial modes), which from now on we
will refer to as ideal vector vortex (VV) beams. In the context of OAM, any
one of the CVV beams with oppoiste spin and OAM states would suffice, as well
as VV beams in the linear polarisation basis. Using such beams as a probe, we
confirm the Choi-–Jamiołkowski isomorphism for classical vectorial light, and
show that the vectorness decay of all initial beams can be predicted from the
decay of an ideal VV beam. The approach is first outlined using CVV beams and
then generalised to other VV beams for adaptability. We illustrate this for
two OAM subspaces, $\ell=\pm 1$ and $\ell=\pm 10$, revealing from this measure
a simple factor for the rate at which one subspace decays relative to the
other. Our work not only offers a simple tool for probing classical
communication channels, but also reveals insights into the decay dynamics of
vectorial OAM light. While we have demonstrated the approach with OAM in the
atmosphere, it can easily be adapted to other mode sets and media, and
likewise to hybrid entangled quantum states.

##  II Concepts

Here we elucidate the concept of channel state duality with nonseparable
vector beams for characterising classical beams through perturbing media. We
first introduce the key concepts of nonseparability in vector beams, vector
beam decay through turbulence and then finally channel state duality. The core
idea is that an ideal vector beam is sufficient to predict the behaviour of
any vector beam, even partially vector, through a channel.

Figure 1:  Example polarisation field profiles of a (a) CVV beam (b) VV beam.
The CVV beam only has nonuniform linear polarisation states while the VV beam
varies between linear and elliptical polarisation states. Both beams have a
VQF of 1. Experimental polarisation fields of vector modes transmitted through
turbulence with varying strengths from left to right, initially encoded in OAM
subspaces (c)-(e) $|\ell|=1$ and (f)-(h) $|\ell|=10$ in the second and third
bottom panels, respectively. The turbulence strengths are $D/r_{0}=0$ (no
turbulence), $D/r_{0}=2.5$ and $D/r_{0}=3.5$ in each column, respectively.

###  II.1 Nonseparable vector modes

To construct an ideal VV beam we require a vectorial superposiiton of spatial
mode and polarization where the nonseparability is maximum. We illustrate the
concept of nonseparability using familiar CVV beams since they are ubiquitous
in a myriad of applications Rosales-Guzmán et al. (2017). Thereafter, we
extend the concept to a family of VV beams by a simple change of basis in the
polarisation components.

CVV beams are natural solutions of the vectorial Helmholtz equation in
cylindrical coordinates. An example of one such beam, is shown in Fig. 1 (a),
having a radially symmetric polarisation field. Such beams are commonly
represented as superpositions of scalar fields coupled to orthogonal circular
polarisation states, i.e.,

$\Psi_{\ell}(\textbf{r})=a\exp(i\ell\phi)\hat{e}_{R}+b\
\exp(-i\alpha)\exp(-i\ell\phi)\hat{e}_{L},$ (1)

where $\textbf{r}=(r,\phi,z=0)$ are the cylindrical coordinates,
$\hat{e}_{R,L}$ are the canonical right and left circular polarisation states
and coupled to spatial components having characteristic azimuthal phase
profiles, $\exp(\pm i\ell\phi)$, associated with light fields carrying an OAM
of $\pm\ell\hbar$ per photon, respectively. Here, the unbounded integer,
$\pm\ell$, is the topological charge. The parameters, $a$, $b$ and $\alpha$
are the relative amplitudes and phases between the the modes in the
superposition. Next, we show that the coupling between the polarisation and
spatial components is reminiscent of quantum entanglement between two
particles.

Figure 2: (a) Illustration of the experimental set up. A Helium Neon (He-Ne)
laser was expanded and collimated onto a spatial light modulater (SLM). On the
SLM, two modes with oppositely charged OAM and distinct gratings frequencies
were encoded on a single hologram as shown in (b). Upon propagation, the modes
separated in path. Since they propagate closely, a D–shaped mirror (DM) as
used to redirect one of them. A half-wave (HW) plate was used to rotate the
polarisation of redirected beam. The modes were subsequently recombined at a
beam-splitter (BS). The resulting mode was imaged to the digital micromirror
device (DMD) where (c) turbulence was encoded in combination with the
detection holograms. Polarisation projections were performed with a linear
polariser. Finally, the resulting mode was propagated to the far field with a
500 mm Fourier lens (L) where an on-axis intensity measurement was performed
with a CCD camera. (d) Example of measurements needed to calculate the
nonseparability, VQF, of vector modes.

It is common to represent the polarisation and spatial components of vector
modes as state vectors using the Dirac notation from quantum mechanics, i.e.,
$\hat{e}_{R(L)}\rightarrow\ket{R(L)}$ and $\exp(\pm
i\ell\phi)\rightarrow\ket{\pm\ell}$, enabling a more compact representation of
Eq. (1) following

$\displaystyle\ket{\Psi_{\ell}}=a\ket{\ell}\ket{R}+b\exp(-i\alpha)\ket{-\ell}\ket{L}.$
(2)

Here the bra-ket notation is used to mark the spatial and polarisation
components making it convenient to express each Degree of Freedom (DoF) as a
unique subsystem analogous to two particle states in quantum mechanics. By
clearly identifying each DoF, we wish to quantify the amount of
nonseparability between them. This can be achieved by using the vector quality
factor (VQF) McLaren et al. (2015); Ndagano et al. (2016), which is an
analogous measure of entanglement based on the concurrence Wootters (2001),
but between the internal DoF of the classical light fields. For the state in
Eq. (2), the VQF, equivalently concurrence, is therefore given by
$\text{VQF}=|ab|$ ranging from $\text{VQF}=0$ for a separable scalar beam
$(a=0\ \text{or}\ b=0)$ to $\text{VQF}=1$ for a nonseparable vector beam
($a=b$); and otherwise partially vector for $0<\text{VQF}<1$. This measure has
been used in a myriad of experiments as a witness for nonseparability in
classical beams Sroor et al. (2018); Otte et al. (2018); Selyem et al. (2019).

Note that such beams are spanned on a four dimensional state-space and
therefore have the general form

$\displaystyle\ket{\Phi_{\ell}}$
$\displaystyle=a\ket{\ell}\ket{R}+b\ket{-\ell}\ket{R}$
$\displaystyle+c\ket{\ell}\ket{L}+d\ket{-\ell}\ket{L}\big{)},$ (3)

with a corresponding degree of nonseparability

$\text{VQF}=|ad-cb|,$ (4)

assuming the coefficients satisfy, $(|a|^{2}+|b|^{2}+|c|^{2}+|d|^{2})=1$. One
can simply recover the previous vector beam for example by setting ($a=b$)
while $cb=0$. This will become crucial when studying the decay of vector beams
since we will always project onto the entire subspace ($\pm\ell$) of spatial
modes we started with.

Next, we show that the coupling between the DoFs of any spatial mode that is
transmitted through a complex medium, using turbulence as an example, can be
determined by that of a maximally nonseparable vector mode by exploiting
channel state duality.

###  II.2 Propagation of vector modes through turbulence and channel state
duality

Figure 3: Measured modal spectrum for the vertical and horizontal polarised
components of vector modes in OAM subspaces corresponding to $\ell=1$ (top
panel) and $\ell=10$ (bottom panel), with (a,d) no turbulence $D/r_{0}=0$ and
increased turbulence strengths of (b,e) $D/r_{0}=2.5$ and (c,f) $D/r_{0}=3.5$.
The solid lines correspond to the theoretical spectrum.

Structured light is known to be perturbed in atmospheric turbulence Cox et al.
(2020). In particular, a vector mode propagating through turbulence
experiences phase dependent fluctuations that have an impact on its transverse
spatial components. Assuming weak irradiance fluctuations approximated by
Kolmogorov theory Andrews and Phillips (2005), the phase variations can be
characterised by the phase structure function,
$D_{\phi}(\mathbf{r}_{1},\mathbf{r}_{2})=6.88\Delta r/r_{0}$, where $r_{0}$ is
the Fried parameter Fried (1965) describing the transverse scale of the
atmospheric distortions and $\Delta r=|\mathbf{r}_{1}-\mathbf{r}_{2}|$ are
relative displacements in the transverse plane. For an optical system with a
diameter (aperture) $D$, we can associate the turbulence strength with the
normalised aperture size, $D/r_{0}$, relating turbulence strength to the
relative transverse distance within which the refractive index is correlated.
For example a large aperture size seeing a smaller Fried parameter ($D>r_{0}$)
experiences more distortions than a smaller aperture seeing a larger Fried
parameter ($D<r_{0}$). In this paper, $D$ approximates the size of the beam.
We show examples of the the effect of turbulence on the polarisation field of
vector modes in Fig. 1 (c)-(h). The polarisation field as well as the spatial
distribution gets deformed with increasing turbulence strength (from left to
right). We investigate how the distortions affect the nonsepability.

Now, since the atmosphere is non-birefringent, only the transverse spatial
components of the mode expressed in Eq. (2) are perturbed and as a consequence
there is modal scattering into adjacent OAM modes Chen and Yang (2016). For
example, an OAM mode corresponding to the state $\ket{\ell}$ traversing a
medium with the channel matrix $\hat{T}=\sum_{m,n}c_{n,m}\ket{m}\bra{n}$,
transfers energy from the initial state into its neighbouring eigenmodes
following the mapping

$\ket{\ell}\xrightarrow{\hat{T}}\sum_{m}c_{\ell,m}\ket{m},$ (5)

where $P(\ell,m)=|c_{\ell,m}|^{2}$ is the conditional probability for the mode
$\ket{\ell}$ to exchange energy with the mode $\ket{m}$. For Kolmogorov
turbulence, the probabilities have been determined analytically Zhang et al.
(2020) and are symmetric about $|\ell|$ for weak turbulence.

With this in mind, we can now describe the state of our CVV mode after it
traverses the turbulent channel. Let us assume that vector mode is initially
in the state

$\displaystyle\ket{\Psi_{\ell}}=1/\sqrt{2}\left(\ket{\ell}\ket{R}+\ket{-\ell}\ket{L}\right).$
(6)

Upon traversing the channel, the OAM modes scatter according to Eq. (5). By
projecting back onto the initial OAM subspace, i.e.,
$\\{\ket{-\ell},\ket{\ell}\\}$, we obtain the state

$\displaystyle\ket{\tilde{\Psi}_{\ell}}$
$\displaystyle=\mathcal{N}\big{(}c_{\ell,\ell}\ket{\ell}\ket{R}+c_{\ell,-\ell}\ket{-\ell}\ket{R}$
$\displaystyle+c_{-\ell,\ell}\ket{\ell}\ket{L}+c_{-\ell,-\ell}\ket{-\ell}\ket{L}\big{)}.$
(7)

Here $\mathcal{N}$ is a normalisation factor satisfying,
$|\braket{\tilde{\Psi}_{\ell}}{\tilde{\Psi}_{\ell}}|^{2}=1$. The VQF becomes

$\mbox{VQF}_{\text{max}}=2|\mathcal{N}|^{2}|c_{\ell,-\ell}c_{-\ell,\ell}-c_{\ell,\ell}c_{-\ell,-\ell}|.$
(8)

By applying the notion of the Choi-–Jamiołkowski isomorphism (channel state
duality), we hypothesis that any other vector mode with initial vectorness of
$\mbox{VQF}_{\text{in}}$ will decay according to the factorisation law,

$\mbox{VQF}_{\text{out}}=\mbox{VQF}_{\text{max}}\times\mbox{VQF}_{\text{in}}$
(9)

where $\mbox{VQF}_{\text{out}}$ is the nonseparability of the state after the
channel. In other words, the decay of the any arbitrary vector beam can be
inferred by simply propagating an ideal vector beam through the channel to
find the scale factor, $\mbox{VQF}_{\text{max}}$. Rather than propagating many
beams through the channel, only one beam has to be passed through to
understand its impact. Remarkably, the channel’s properties are determined by
its interaction with a maximally nonseparable vector mode. This means that by
knowing how a maximally nonseparable vector mode propagates through
turbulence, it is possible to predict how the nonseparability of any other
arbitrary superposition state evolves. Equation (9) also predicts that the
trend is linear and an intercept at zero, with the decay of the CVV beam
returning the slope ($\mbox{VQF}_{\text{max}}$).

This intriguing property, can be understood from the perspective of quantum
mechanics. Firstly, the channel weights,
$|\braket{\pm\ell,}{\hat{T}}{\ell}|^{2}$, are imprinted on the input
nonseparable state mapping the channel onto a pure state, as a consequence of
the Choi–Jamiołkowski isomorphism Choi (1975); Jamiołkowski (1972).
Equivalently, a vector mode traversing a noisy channel has similar properties
owing to the nonseparability of the spatial and polarisation components. This
means that the final VQF of any partial nonseparable mode can be determined by
that of a maximally nonseparable vector mode.

Finally, although we have used the CVV beam as a well-known example, in
general the polarisation and spatial components can be in any basis. For
convenience going forward and in the experiment, we will convert to the
horizontal ($\ket{H}$) and vertical ($\ket{V}$) polarisation basis while the
spatial profiles are defined in the Laguerre-Gaussian (LG) basis. As a result,
we no longer have a CVV beam since there is no cylindrical symmetry in the
polarisation field. An example of our VV beam is shown in Fig. 1 (b), where
the polarisation states across the tranverse plane are mixtures of linear of
elliptical states. Such a beam is represented by the state,

$\displaystyle\ket{\Psi_{\ell}}$
$\displaystyle=a\ket{\ell}\ket{H}+b\ket{-\ell}\ket{V},$ (10)

with the same VQF = 1 as that of Eq. (2). This serves to make it clear than
any ideal VV is sufficient for the test.

##  III Methods

###  III.1 Experimental set-up

We describe the generation and detection scheme illustrated in Fig. 2 (a). We
used a Helium-Neon (He-Ne) laser with a central wavelength of 633 nm and
collimated Gaussian field profile. We modulated the laser beam using a Phase
only Holo-Eye Pluto spatial light modulate (SLM). To obtain the states
$\ket{\Psi}_{\ell=1,10}$, we encoded the multiplexed holograms Rosales-Guzmán
et al. (2017) with Laguerre Gaussian (LG) modes of charges $\ell=-1(-10)$ and
$\ell=1(10)$ and subsequently separated them in path using a D shaped mirror.
The amplitudes and phases of each mode were encoded using the Arrizon
technique for complete amplitude and phase control Arrizón et al. (2007). An
example of one of the holograms is shown in Fig. 2 (b).

Before interfering the two beams at the BS, we rotated the polarisation of the
reflected beam from the D–shape mirror by $45^{\circ}$ using a half wave-
plate. This converted the polarisation from H to V. The two beams now had
orthogonal polarisations. After combining the two beams, the resulting vector
mode was transmitted to the digital micro-mirror device (DMD). On the DMD, we
encoded Kolmogorov turbulence phase screens following Zhao et al. (2012) in
combination with the detection holograms necessary for the VQF measurements.
An example of one of the detection holograms is shown in Fig. 2 (c) as a
combination of the detection mode and turbulence phase screen, resulting in a
noisy detection hologram that has both the perturbation from turbulence and
projection mode. The VQF projection holograms, with no turbulence, had phase
profiles shown in the first row of Fig. 2 (d), shown for the $\ell=\pm 1$
subspace.

Lastly we used a polariser to project onto the H and V polarisation modes
after the DMD. The resulting field was then propagated to the far field using
a Fourier lens (L) and an on axis measurement of the intensity was recorded,
providing the modal overlap of the input state, the simulated turbulence and
the detection mode Flamm et al. (2012). For each measurement we prepared up to
30 instances of each turbulence strength ranging from $D/r_{o}$=$0$ to $3.5$
in steps of $D/r_{o}=0.5$. An example of measurements for intensities $I_{uv}$
is shown in Fig. 2 (d) for a perfect vector mode. The columns correspond to
the spatial projections while the rows correspond to the polarisation
measurements. Next we show how the VQF (nonseparability) is measured.

###  III.2 Vector quality factor measurement

We follow the procedure outlined in ref. McLaren et al. (2015) to measure the
nonseparability of vector modes. The VQF is given by

$\text{VQF}=\sqrt{1-\sum^{3}_{i}\braket{\sigma_{i}}^{2}},$ (11)

where the expectation values of the Pauli matrices $\langle\sigma_{i}\rangle$
can be obtained from

$\displaystyle\langle\sigma_{1}\rangle=I_{13}+I_{23}-(I_{15}+I_{25}),$ (12)
$\displaystyle\langle\sigma_{2}\rangle=I_{14}+I_{24}-(I_{16}+I_{26}),$ (13)
$\displaystyle\langle\sigma_{3}\rangle=I_{11}+I_{21}-(I_{12}+I_{22}).$ (14)

The detection probabilities $I_{uv}$, $u=\\{1,2\\}$ , $v=\\{1,2,...,6\\}$ are
determined from six identical projections of different polarisation basis
states, namely horizontal and vertical polarisations. The projections are
performed by inserting a polariser, set to $0^{\circ}$ and $90^{\circ}$ for
the horizontal and vertical polarisation projections, respectively. The six
spatial measurements consist of projections onto OAM states $\ket{\pm\ell}$
and their four superpositions
$\ket{\theta}=\ket{-\ell}+e^{i\frac{\theta}{|\ell|}}\ket{\ell}$ with
$\theta=0,\frac{\pi}{2},\pi$ and $\frac{3\pi}{2}$. The spatial projections
were encoded as binary holograms onto the digital micro-mirror device using
the method in Lee (1979) tailored for amplitude only devices. Finally, the on-
axis intensity, due to each projection is measured in the focal plane of a
Fourier lens by a CCD camera, in the first diffraction order.

##  IV Results

Figure 4: (a) Experimental results (points) and theoretical prediction (lines)
for the degree of nonsepability of vector modes in the subspaces, $\ell=1$ and
$10$, with increasing turbulence strength. Experimental output VQF with
respect to the known input VQF under the effect of turbulence strengths of (b)
$D/r_{0}=2.5$ and (c) $D/r_{0}=3.5$. In the absence of perturbations, the
output VQF maps onto the diagonal (“No perturbation”) line. The circles are
for vector modes in the subspace of $\ell=1$ and squares are for $\ell=10$,
while the lines are the theoretical prediction based on the isomorphism. The
horizontal error-bars are smaller than the points. Each data point was
obtained from 30 realisations of the same turbulence strength.

###  IV.1 Propagation of vector modes through turbulence

We first generated vector modes ($\ket{\Psi_{\ell}}$) and measured their
polarisations using Stokes polirometry. The polarisations profiles are shown
in Fig. 1 (c)-(e) and (f)-(h) for $\ket{\Psi_{1}}$ and $\ket{\Psi_{10}}$,
respectively. In each panel, the profiles are shown for increasing turbulence
strengths $D=0,2.5$ and $D=3.5$, from left to right. The intensity profile is
shown to deteriorate in each instance confirming the presence of distortions
in the transverse plane of the fields.

Next, we measured the modal spectrum of each polarisation component averaged
over 30 instances of the same turbulence strength. The results are shown in
Fig 3 (a)-(c) for $\ket{\Psi_{\ell=1}}$ and in Fig 3 (d)-(f) for
$\ket{\Psi_{\ell=10}}$. For each $\ell$ dependent mode, the turbulence
strength was $D/r_{0}$ = 0 (a , d) $D/r_{0}$=2.5 (b , e) and $D/r_{0}$=3.5 (c,
f). In each plot, the distribution on the right corresponds the horizontally
polarised mode (blue) while the distribution on the left (red) corresponds to
the vertically polarised mode. As expected, the mode distribution is symmetric
about $|\ell|$, consistent with the theoretical distribution shown as lines
Zhai et al. (2020). We measured the width of each distribution as 2 times the
standard deviation using the formula

$\Delta\ell=2\sqrt{\frac{\sum_{m}|m-\overline{\ell}|P(\ell,m)}{\sum_{m}P(\ell,m)}},$
(15)

where $\overline{\ell}$ is the mean OAM in the field. For $D/r_{0}=2.5$ we
measured a width of $\Delta\ell=3.18$ and $\Delta\ell=6.64$ for
$\ket{\Psi_{\ell=1}}$ and $\ket{\Psi_{10}}$, respectively, averaged over both
polarisation components of the beams. With an increased turbulence strength,
$D/r_{0}=3.5$, we measured a width of $\Delta\ell=4.44$ and $\Delta\ell=8.79$.
In both cases the OAM width of $\ell=10$ vector modes is higher than that of
$\ell=1$ vector modes (2 times), showing that higher order OAM modes spread
farther than lower order OAM modes, consistent with theory Chen and Yang
(2016).

While this reveals information about how the spatial components are perturbed,
we now investigate how the polarisation in tandem with the spatial components
are affected, testing our isomorphism hypothesis.

To illustrate the effect of the mode scattering on the nonseparability, we
show the theoretical (lines) and experimental (points) of VQF values measured
for $|\ell|=1,10$ subspaces as a function of turbulence strength in Fig. 4
(a). The VQF of the $\ell=10$ modes decay at lower rate than the VQF of the
$\ell=1$ modes, analogous to the decay of entangled photons through a single
sided channel of turbulence Ibrahim et al. (2013). This can be explained by
the larger mode separation between higher OAM suspaces enabling for less
crosstalk between highly separated Malik et al. (2012).

Table 1: Analysis of the different OAM subspace through turbulence. The values in brackets correspond to the theoretical values. Here the gradients, $m_{\ell}$, correspond to $\mbox{VQF}_{\text{max}}$. $D/r_{0}$ | $\Delta\ell\ (\ell=1)$ | $\Delta\ell\ (\ell=10)$ | $m_{1}$ | $R^{2}_{1}$ | $m_{10}$ | $R^{2}_{10}$
---|---|---|---|---|---|---
0 | 0.05 (0) | 0.84 (0) | - | - | - | -
2.5 | 3.18 (4.1) | 6.64 (9) | 0.40 | 0.97 | 0.94 | 0.99
3.5 | 4.44 (5.59) | 8.79 (11.94) | 0.21 | 0.97 | 0.92 | 0.98

While the above analysis was performed on input modes with a high
nonseparability ($VQF\approx 1$), next we evaluated how vector modes with a
varying degree of nonseparability also decay under the same turbulence
strength. We demonstrate this to confirm the channel state duality inherent in
vector modes propagated through a perturbation channel acting on the spatial
DoF. The results are shown in Fig. 4 (b) and (c) under turbulence conditions
of $D/r_{o}=2.5$ and $3.5$, respectively. This was done for subspaces
$\ell=\pm 1$ (circles) and $\ell=\pm 10$ (squares).

The error-bars for the x-axis are smaller than the points. To control the VQF
of the input mode, we adjusted the grating depth of the hologram corresponding
to the $\ell=-1(-10)$ mode. The plots show that the output VQF, i.e.,
$\mbox{VQF}_{\text{out}}$, has a linear relation to the input VQF where the
line fitted through each data set has a gradient, $m_{\ell}$ or equivalently
$\mbox{VQF}_{\text{max}}$, that is equivalent to the VQF of a maximally
nonseparable vector mode transmitted through the same turbulence. We show the
gradient of each line in Table 1 as well as the goodness of fit, which are
above $R^{2}=0.96$. A perfect fit would result in $R^{2}=1$. As shown, the
gradients of the $\ell=10$ modes is higher than that of $\ell=1$, owing to the
higher crosstalk in the $\ell=1$ subspace as demonstrated earlier. For
$D/r_{o}=3.5$, we find that the $\ell=10$ subspace, the gradient can be 4
times larger in comparison to the $\ell=1$ subspace. Since the gradient
indicates the maximum VQF that a vector mode with an input nonseparability of
$\mbox{VQF}_{\text{in}}\approx 1$ can obtain, after propagating through the
channel, all partially nonseparable vector modes within the same subspace are
bounded on the interval $\mbox{VQF}_{\text{out}}\in[0,m_{\ell}]$ consistent
with the factorisation law Konrad et al. (2008) for single sided channels
indicating that vector modes posses the ability to probe channel state duality
of noisy channels. Indeed, our results show that the decay in nonseparability
of any vector mode, in the same subspace decays through the medium according
to the relation $\mbox{VQF}_{\text{out}}=\mbox{VQF}_{\text{in}}\times
m_{\ell}=\mbox{VQF}_{\text{in}}\times\mbox{VQF}_{\text{max}}$ where
$\mbox{VQF}_{\text{max}}=\mbox{VQF}(\hat{T}\ket{\Psi}_{\ell})$ is the VQF of a
vector mode, $\ket{\Psi}_{\ell}$, after traversing the channel, $\hat{T}$, and
having an initial VQF of 1 while $\mbox{VQF}_{\text{in}}$ is the VQF of the
mode we wish to characterise after the channel. The isomorphism holds for
vectorial classical light and can be used to probe classical channels.

##  V Discussion

Vector beams possess nonseparable coupling between their polarisation and
spatial components and exhibiting correlations similar to entangled pairs of
photons. In this paper, we used this fact to study the decay of vector beams
in atmospheric turbulence both qualitatively and quantitatively by invoking
properties such as channel state duality Choi (1975); Jamiołkowski (1972) and
the factorisation law Konrad et al. (2008). These features are unique to
quantum entangled states, and are commonly used for channel (medium)
characterisation Malik et al. (2010). Our results confirm that the
nonseparability of any other partially ($\text{VQF}<1$) nonseparable vector
mode is purely determined by that of a maximally ($\text{VQF}\approx 1$)
nonseparable vector mode experiencing the same turbulence within the same
subspace of optical modes. Interestingly, this does not only limit the method
to characterisation purposes but creates the possibility of using vector modes
as a means to overcoming turbulence, for example by selecting higher order
modes (e.g $\ell=10$ as apposed to $\ell=1$), or for unscrambling complex
aberrations shown to be feasible for high dimensional quantum states Valencia
et al. (2020). The extension of the latter to classical beams could be used as
an additional tool for adaptive optics since vector modes can carry
information related to the channel/medium. While we demonstrated this method
for turbulence, it can in principle be extended to various scenarios where
optical aberrations are encountered, for example, arising from overheating
optical elements in high power regimes or imperfections in optical elements.

##  VI Conclusion

In summary, we exploited the concept of channel state duality analogously to
characterise the evolution of various vector/scalar modes through turbulence,
demonstrating that the evolution of vector beams can be used to study how
various spatial modes decay through turbulence. We showed this for two
subspaces, $\ell=1$ and $\ell=10$, with our results demonstrating that higher
order OAM vector modes decay rapidly while also maintaining a high
nonseparability. Our work is integral to the development of alternative
methods for characterising optical beams by borrowing principles from quantum
mechanics, with possible applications in various scenarios where complex
perturbations are encountered.

## Appendix

###  VI.1 Scattering probability of OAM in turbulence

The detection probability of an OAM beam propagated through turbulence can
computed from

$P(l,m)=\iint
C_{\psi}(r,\Delta\theta,z)rdr\times\frac{\exp(-im\Delta\theta)}{2\pi}d\Delta\theta,$
(16)

where $\ell$ is the input OAM index of the beam, and $m$ is the index for the
scattered mode, while $C_{\psi}\left(r,\Delta\theta,z\right)$ is the
rotational coherence function defined in the cylindrical coordinates,
$(r,\theta,z)$ defined as Paterson (2005)

$C_{\psi}(r,\Delta\theta,z)=\braket{\psi^{*}(r,0,z)\psi(r,\Delta\theta,z)}.$
(17)

Here $\psi(r,\theta,z)=u(r,\theta,z)\exp(i\phi(r,\theta))$ is the beam profile
after propagating a distance $z$ with an initial profile of $u(r,\theta,0)$ at
the waist plane and $\exp(i\phi(r,\theta))$ is the accumulated phase according
to the Rytov approximation.

For LG beams, the integral in Eq. (16) has been solved analytically, yielding
the in the expression Zhai et al. (2020)

$\displaystyle P(l,m)$
$\displaystyle=(\frac{1}{t})^{|\ell|+1}\left(\frac{t-1}{t+1}\right)^{n}\sum_{k=0}^{|\ell|}\binom{|\ell|+n}{k}$
$\displaystyle\times\binom{2|\ell|-k}{\ell}\binom{4t}{(t-1)^{2}}^{k-|\ell|}$
(18)

where $n=|\ell-m|$, $t=\sqrt{1+\zeta}$ while $\zeta=3.44\times
2^{2/3}(w_{0}/r_{0})^{2}$ where $w_{0}$ is the is the waist size of the
Gaussian argument in the LG mode and $r_{0}$ is the Fried parameter. We re-
scaled the Gaussian argument, i.e. $w_{\ell}=w_{0}/\sqrt{|\ell|+1}$, so that
each OAM mode has the same diameter, $D:=\sqrt{8}w_{\ell}$ Andrews and
Phillips (2005) and turbulence strength, $D/r_{0}$.

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