figure table

# Robust structured light in atmospheric turbulence

Asher Klug  School of Physics, University of the Witwatersrand, Private Bag 3,
Wits 2050, South Africa  Cade Peters  School of Physics, University of the
Witwatersrand, Private Bag 3, Wits 2050, South Africa  Andrew Forbes
<EMAIL_ADDRESS>School of Physics, University of the Witwatersrand,
Private Bag 3, Wits 2050, South Africa<EMAIL_ADDRESS>
###### Abstract

Structured light is routinely used in free space optical communication
channels, both classical and quantum, where information is encoded in the
spatial structure of the mode for increased bandwidth. Unlike polarisation,
the spatial structure of light is perturbed through such channels by
atmospheric turbulence, and consequently, much attention has focused on
whether one mode type is more robust than another, but with seemingly
inconclusive and contradictory results. Both real-world and experimentally
simulated turbulence conditions have revealed that free-space structured light
modes are perturbed in some manner by turbulence, resulting in both amplitude
and phase distortions. Here, we present complex forms of structured light
which are invariant under propagation through the atmosphere: the true
eigenmodes of atmospheric turbulence. We provide a theoretical procedure for
obtaining these eigenmodes and confirm their invariance both numerically and
experimentally. Although we have demonstrated the approach on atmospheric
turbulence, its generality allows it to be extended to other channels too,
such as underwater and in optical fibre.

##  I Introduction

Free-space transmission of electromagnetic waves is crucial in many diverse
applications, including sensing, detection and ranging, defence and
communication, and extends over distances from the long (Earth monitoring) to
the short (WiFi and LiFi). Lately there has been a resurgence of interest in
free-space optical links trichili2019communicating ; trichili2020roadmap ,
driven in part by the need for increased communication bandwidths Richardson1
; richardson2013space , with the potential to bridge the digital divide in a
manner that is license-free lavery2018tackling . Here the spatial modes of
light have come to the fore, for so-called space division multiplexing
li2014space  and mode division multiplexing Berdague1982A , where the spatial
structure of light is used as an encoding degree of freedom. This in turn has
fuelled interest in structured light forbes2021structured ;
forbes2020structured , where light is tailored in all its degrees of freedom,
including amplitude, phase and polarisation, enabled by a modern structured
light toolkit shen2021rays .

A commonly used form of structured light is that of beams carrying orbital
angular momentum (OAM), where the phase spirals around the path of propagation
azimuthally padgett2017orbital . These modes provide a (theoretically)
infinite and easily realised alphabet for encoding information Gibson2004 ;
wang2012terabit  and have been used extensively in optical communication (see
Refs. Willner2015 ; willner2021orbital  for good reviews). Vectorial
combinations of such beams create inhomogeneous polarisation structures
zhan2009cylindrical ; rosales2018review ; otte2018polarization  and too have
found applications in free-space links nape2021revealing ; Milione2015d ;
Sit2017 . Although these structured light fields hold tremendous potential for
free-space optical communication, they are distorted by atmospheric turbulence
as a phase perturbation in the near-field and an amplitude, phase and
polarisation perturbation in the far-field nape2021revealing . This modal
scattering induced cross-talk decreases the information capacity of classical
atmospheric transmission channels JaimeA.Anguita2008 ; ren2016experimental ;
krenn2014communication ; zhao2016experimental ; rodenburg2012a ;
krenn2016twisted ; zhang2020mode ; malik2012influence ; chen2016changes ;
tyler2009influence  while reducing the degree of entanglement in quantum links
Paterson2005 ; jha2010c ; Tyler2009 ; oamturb ; gopaul2007effect ;
zhang2016experimentally ; pors2011transport ; goyal2016effect ;
leonhard2015universal ; sorelli2019entanglement . Mitigating this remains an
open challenge that is intensely studied.

Arguments have been put forward for one mode family being more robust than
another, with studies covering Bessel-Gaussian mphuthi2018bessel ;
mphuthi2019free ; lukin2014mean ; bao2009propagation ; zhu2008propagation ;
nelson2014propagation ; ahmed2016mode ; cheng2016channel ; doster2016laguerre
; watkins2020experimental ; vetter2019realization ; yuan2017beam , Hermite-
Gaussian cox2019hglg ; ndagano2017c ; Restuccia2016 ; ndagano2017comparing ,
Laguerre-Gaussian Trichili2016 ; zhao2015capacity ; zhou2019using ;
xie2016experimental ; li2017power  and Ince-Gaussian krenn2019turbulence
beams, with mixed and contradictory results. In the context of OAM, since the
atmosphere itself can be thought of giving or taking OAM from the beam, it has
been shown theoretically and experimentally that atmospheric turbulence
distortions are independent of the original OAM mode klug2021orbital , all
susceptible to the deleterious effects of atmospheric turbulence, and indeed
OAM has been suggested as not the ideal modal carrier through turbulence
miller2017better . Vectorial structured light has been suggested to improve
resilience because of the invariance of the polarisation degree of freedom,
but numerous studies in turbulence cox2020structured ; gu2009scintillation ;
cheng2009propagation ; cai2008average ; ji2010propagation ;
wang2008propagation ; Cox:16 ; lochab2018designer  have been inconclusive,
with some reporting that the vectorial structure is stable lochab2018designer
; gu2009scintillation ; cheng2009propagation , and others not cai2008average ;
ji2010propagation ; wang2008propagation ; Cox:16 ; Ndagano2017 . Careful
inspection of the studies that report vectorial robustness in noisy channels
reveal that the distances propagated were short and the strength of
perturbation low, mimicking a phase-only near-field effect where indeed little
change is expected, and hence these are not true tests for robustness or
invariance. Studies that claim enhanced resilience of vector modes over
distances comparable to the Rayleigh length lochab2018designer ;
gu2009scintillation  have used the variance in the field’s intensity as a
measure, a quantity that one would expect to be robust due to the fact that
each polarisation component behaves independently and so will have a low
covariance. This failing of structured light in turbulence has led to numerous
correction techniques, including novel encoding/decoding methods
zhu2019compensation , modal diversity as an effective error-reduction scheme
cox2018modal , traditional adaptive optics for pre- and post-correction
tyson2002bit ; zhao2012aberration ; ren2014adaptive  as well as vectorial
adaptive tools he2020vectorial , iterative routines li2017gerchberg  and deep
learning models liu2019deep .

Figure 1: Propagation through turbulence. Most common forms of structured
light (such as Laguerre-Gaussian modes) become distorted when propagating
through the atmosphere due to the effects of atmospheric turbulence. Compare
this to an eigenmode of atmospheric turbulence, which remains unchanged when
propagating through the same channel.

Here we present a class of structured light whose entire structure in
amplitude and phase remains invariant as they propagate through a turbulent
free-space channel. We deploy an operator approach to find the eigenmodes of
atmospheric turbulence, a significant departure from prior phenomenological
approaches. Unlike other spatial modes, these exotically structured eigenmodes
need no corrective procedures and are naturally devoid of deleterious effects
such as modal crosstalk. Moreover, they are valid over any path length in the
medium so long as the medium conditions remain static over the time frame of
the beam transport, always true for atmospheric turbulence (which typically
changes at the Greenwood frequency of 100s of Hz, much slower than the speed
of light). We demonstrate this invariance numerically and confirm it
experimentally with a laboratory simulated long path comprising weak, medium
and strong turbulence, implemented using multiple turbulent phase screens
along the propagation path. Our approach offers a new pathway for exploiting
structured light in turbulence, and can be easily extended to arbitrary noisy
channels whose characteristics are known.

Figure 2: The unit cell. The first turbulent screen is placed at the beginning
of the channel, at $z=0$, with subsequent screens placed a distance $\Delta
z=L/N$ away from the prior, where $N$ is the number of turbulent phase screens
used. Each phase screen and distance form a unit cell, the first highlighted
in blue, forming $N$ unit cells over the complete path length of $z=L$. The
operator for each unit cell, $\mathcal{T}$, is identical, so we need only
consider the first unit cell. The initial plane is discretised into pixels
with side length $\delta$ and turbulence is simulated with a strength
characterised by the ratio $D/r_{0}$, where $D$ is the aperture of the
inscribed circle and $r_{0}$ is Fried’s parameter. The operator describes the
action of an imprinted turbulent phase on the beam, followed by vacuum
propagation over a distance $\Delta z$.

##  II The eigenmodes of turbulence

The concept we tackle here is illustrated in Fig. 1. Structured light is
typically distorted after propagation through free-space due to atmospheric
turbulence. In contrast, the eigenmodes of turbulence are complex forms of
structured light that are invariant to the channel, emerging distortion free.
To solve for these eigenmodes, we use the multiple screen approximation to
Fig. 1 as shown in Fig. 2. Here we outline the theory and its implications,
before moving on to numerical and experimental confirmation.

###  II.1 Theory

Figure 3: Eigenmodes of turbulence. The numerically calculated eigenmodes of
turbulence, showing the first five modes (columns) as a function of turbulence
strength (rows). The insets show the phase profile. All eigenmode were
calculated for a total propagation path of 100 m through weak, medium and
strong turbulence as defined by the Rytov variance ($\sigma^{2}_{R}$) and
Fried parameter ($r_{0}$). The first two rows show eigenmodes of weak
turbulence with $\sigma^{2}_{R}=0.5$ and $r_{0}=1.8\text{ mm}$. The next two
rows show eigenmodes of medium turbulence with $\sigma^{2}_{R}=1$ and
$r_{0}=1.2\text{ mm}$. The last row shows eigenmodes of strong turbulence with
$\sigma^{2}_{R}=1.5$ and $r_{0}=0.91\text{ mm}$.

The effects of turbulence are mathematically captured in the stochastic
refractive index $n=1+\delta n$, where $\delta n$ is the random variation in
the refractive index of the Earth’s atmosphere. It is assumed that $\delta n$
has a zero mean value, i.e. $\left<\delta n\right>=0$, and that the variation
is small, so $|\delta n|\ll 1$. The introduction of this varying term produces
the stochastic paraxial Helmholtz equation for a field $\text{V}(x,y,z)$

$\left(\nabla^{2}_{t}+2ik\partial_{z}+2k^{2}\delta n\right)\text{V}=0,$ (1)

where $\nabla^{2}_{t}$ is the transverse Laplacian and $k=2\pi/\lambda$ is the
wavenumber for wavelength $\lambda$. Equation 1 can be solved numerically
according to the split-step method schmidt2010numerical , illustrated in Fig.
2. Multiple random phase screens are placed at equal distances along the
beam’s propagation path. Importantly, each screen is in the weak turbulence
limit and contributes a random phase $\Theta_{j}$, where $j$ labels the $j$th
screen, so that a single screen approximation is valid, but the sum of many
such screens can lead to medium or even strong turbulence.

We introduce an operator approach to the problem. We exploit the fact that the
path is subdivided into identical units, each containing such a single screen
and a zero turbulence propagation path of length $\Delta z$. We realise that
in the language of operators, the action of each unit $\hat{\mathcal{U}}$ on
some field V is given by the product of the operators
$\hat{\mathcal{P}}\hat{\mathcal{T}}$, where $\hat{\mathcal{P}}$ and
$\hat{\mathcal{T}}$ refer to free-space propagation and turbulence
respectively. The action of $\hat{\mathcal{U}}$ on a field V is given by the
Huygen-Fresnel integral with a turbulent phase factor

$\hat{\mathcal{U}}\text{V}\equiv\int\differential[2]{\mathbf{x}}\;g(\mathbf{x},\mathbf{x}^{\prime};\Delta
z)\exp(i\Theta)\text{V}(\mathbf{x},z=0),$ (2)

where

$g(\mathbf{x},\mathbf{x}^{\prime};\Delta z)=\frac{1}{i\lambda\Delta
z}\exp\left(\frac{i\pi}{\lambda\Delta
z}\norm{\mathbf{x}-\mathbf{x}^{\prime}}^{2}\right)$ (3)

is the paraxial free space Green’s function and $\mathbf{x}$,
$\mathbf{x}^{\prime}$ are the two-dimensional coordinates of the initial and
final planes, respectively. We then discretize $\mathbf{x}$ and
$\mathbf{x}^{\prime}$ into grids of $N\times N$ points. The coordinates are
labelled $\mathbf{x}\equiv(x_{\alpha},y_{\beta})$ and
$\mathbf{x}^{\prime}\equiv(x_{\mu},y_{\nu})$, so that $\hat{\mathcal{U}}$ is
given by

$\begin{gathered}\mathcal{U}_{\mu\nu\alpha\beta}=\frac{1}{i\lambda\Delta
z}\exp\left(\frac{i\pi}{\lambda\Delta
z}\left(x_{\mu}-x_{\alpha}\right)^{2}\right)\\\
\times\exp\left(\frac{i\pi}{\lambda\Delta
z}\left(y_{\nu}-y_{\beta}\right)^{2}\right)\exp\left(i\Theta(x_{\alpha},y_{\beta})\right).\end{gathered}$
(4)

An eigenmode E is then a solution to the tensor eigenvalue equation

$\gamma_{n}\text{E}^{n}_{\mu\nu}=\mathcal{U}_{\mu\nu\alpha\beta}\text{E}^{n}_{\alpha\beta},$
(5)

where $\gamma_{n}$ is the eigenvalue of the nth eigenmode. Repeated indices
are implicitly summed over and
$\text{E}_{\mu\nu}\equiv\text{E}(x_{\mu},y_{\nu})$. To convert the above
tensor equation into the usual matrix-vector form, we specify a mapping $\rho$
that acts on the indices $(\alpha,\beta)$ and $(\mu,\nu)$ and “counts” them,
first by columns and then by rows, such that $\rho(1,1)=1$, …, $\rho(N,1)=N$,
$\rho(1,2)=N+1$ up to $\rho(N,N)=N^{2}$. This mapping lets us rewrite Eq. 5 as

$\gamma_{n}\text{E}^{n}_{i}=\mathcal{U}_{ij}\text{E}^{n}_{j},$ (6)

since $\rho(\alpha,\beta)=j$ and $\rho(\mu,\nu)=i$. This equation can be
routinely solved using numeric methods to find the eigenmodes of the unit cell
operator. The action of the full channel is then described by the product
$\hat{\mathcal{U}}_{n}...\hat{\mathcal{U}}_{1}$ of repeated unit cells, and as
per the definition of eigenmodes, they remain invariant regardless of the
number of operators applied.

Turbulence is a stochastic process in which the refractive index of the
Earth’s atmosphere varies according to well-known statistics, having zero mean
and some non-zero variance. To see the impact of averaging over many different
instances of turbulence on the robustness of modes, we return to the the
Helmholtz equation but this time in the more general non-paxarial form

$\left(\nabla^{2}+k^{2}\right)\text{V}=-2k^{2}\delta n\text{V},$ (7)

which has the solution

$\text{V}(\mathbf{r}^{\prime})=2k^{2}\int\differential[3]{\mathbf{r}}\;G(\mathbf{r},\mathbf{r}^{\prime})\text{V}(\mathbf{r})\delta
n(\mathbf{r}),$ (8)

with
$G(\mathbf{r},\mathbf{r}^{\prime})=\exp(ik\norm{\mathbf{r}-\mathbf{r}^{\prime}})/4\pi\norm{\mathbf{r}-\mathbf{r}^{\prime}}$
being the free-space Green’s function and $\mathbf{r}=(\mathbf{x},z)$. Taking
the ensemble average and using the result that $\left<\delta
n\text{V}\right>=\mathcal{A}\left<\text{V}\right>$ andrews2005laser , we find

$\left<\text{V}(\mathbf{r}^{\prime})\right>=2k^{2}\mathcal{A}\int\differential[3]{\mathbf{r}}\;G(\mathbf{r},\mathbf{r}^{\prime})\left<\text{V}(\mathbf{r})\right>,$
(9)

where the constant $\mathcal{A}$ is related to the covariance of the
refractive index fluctuations.

We recognize that Eq. 9 is identical to the usual, zero-turbulence Fresnel
integral, up to a constant. Therefore, the averaged eigenmodes should be
solutions to the free-space, no turbulence, case. In other words, if the
channel involves some form of averaging, say at the detector, then the best
mode set in this case is identically the traditional free-space modes in
various geometries: Hermite-Gaussian, Laguerre-Gaussian and so on.

###  II.2 Numerical simulation

To validate the theory, we first use the operator to calculate the eigenmodes,
and then numerically propagate them through a thick medium of atmospheric
turbulence using the split-step approach illustrated in Fig. 2. Throughout the
text, for clarity and brevity, we show only the low order eigenmodes and use
the OAM modes as our point of comparison. Examples of the intensity and phase
of the eigenmodes are shown in Fig. 3. Here the first five eigenmode solutions
are shown in the left column, increasing from left to right, with the rows
corresponding to the turbulence strength, increasing from top to bottom.
Although these are complex forms of structured light, as eigenmodes to
turbulence they should be invariant after propagation through a turbulent
atmosphere. To test this, we propagate OAM carrying Laguerre-Gaussian (LG)
modes and the eigenmodes through various scenarios of turbulence over a 100 m
path length, with the results shown in Fig. 4. We see that while the OAM modes
are distorted, the eigenmodes are robust. This can be quantified by performing
a modal analysis pinnell2020modal  at the end of the turbulent channel, as
would be the case in optical communication at the receiver. In Fig. 5 we see
that while the crosstalk is substantial for OAM modes when propagated through
turbulence, evident from the many off-diagonal terms, the eigenmode crosstalk
matrix remains diagonal after the same channel, for minimal crosstalk.

Figure 4: Invariance of the eigenmodes. The eigenmodes (left) and LG modes
(right) after propagation through weak, medium and strong turbulence through a
channel equivalent to propagating over a distance of 100 m. The insets shows
the modes before experiencing turbulence. The numerical simulations used the
split step method with three unit cells each consisting of a turbulence screen
with a given $r_{0}$ followed by $33.33\text{ m}$ of propagation. Weak
turbulence was characterised by $\sigma^{2}_{R}=0.5$ and $r_{0}=1.8\text{
mm}$, medium turbulence by $\sigma^{2}_{R}=1$ and $r_{0}=1.2\text{ mm}$ and
strong turbulence by $\sigma^{2}_{R}=1.5$ and $r_{0}=0.91\text{ mm}$

Figure 5: Cross talk free transmission. Cross talk matrices for OAM modes
$\ell\in[0,4]$ (top) and eigenmodes (bottom) with insets showing the intensity
of the beams. The eigenmodes are unchanged and remain orthogonal, whereas the
OAM modes scatter into each other. Turbulence results shown for $D/r_{0}=2$
with a total path length of 100 m and a beam waist parameter for the OAM beams
of $w_{0}=6.67\text{ mm}$.

It is instructive to consider the evolution from the free-space modes to the
eigenmode structure as turbulence is steadily increased, shown graphically in
Fig. 6. When there is no turbulence ($D/r_{0}=0$) the operator correctly
returns the free-space modes, with the inset showing the first order solution
- the Gaussian mode. As turbulence increases so the eigenmode structure
changes in amplitude and phase, with insets showing the new first order
eigenmodes. The correlation of the new eigenmodes to the initial free-space
modes are plotted versus turbulence strength. Initially the operator returns
the true free-space modes (correlation of 1), but as turbulence increases so
the deviation of the eigenmodes from the free-space modes increases, resulting
in a decreased correlation. For higher-order modes (shown up to order 6), the
deviation is so great as to make the correlation close to zero: there is
little similarity remaining between the free-space mode and the eigenmode of
turbulence. This likely explains why no free-space modal family has been found
to be robust to turbulence - they are just “too far” in a modal sense from
what is required.

Figure 6: Evolving robust structure. The correlation (overlap integral) for
four orders (1, 4, 5 and 6) of eigenmodes with their corresponding vacuum
counterparts is shown for a range of turbulence strengths, characterised by
$D/r_{0}$. Insets show intensity patterns for the first order ($n=1$)
eigenmode at various turbulence strengths. Figure 7: Modal averaging. The
averaged fidelity as a function of turbulence strength for vacuum and two
examples of eigenmodes (set 1 and set 2). The eigenmodes are calculated for a
particular instance of turbulence but used in conditions that differ from
this. The fidelity is calculated as the average value of the elements along
the diagonal of their $4\times 4$ cross talk matrices. While the vacuum modes
are perfectly orthogonal in the absence of turbulence (fidelity of 1), their
fidelity decreases for increasing turbulence strength. The turbulent
eigenmodes are not orthogonal in the absence of turbulence, and their fidelity
is consistently worse than the vacuum modes under this averaging effect.

In contrast, if the conditions are varying faster than the eigenmodes can be
altered, and the receiver averages the output from the channel, then as
predicted by our theory, the free-space modes outperform the eigenmodes, as
shown in Fig. 7. Here, the robustness of the performance of the eigenmodes is
examined when the turbulence conditions are changed and the detection
averaged. The many different instances of turbulence exclude the phase screens
used to form the modes initially. Four beams (mode orders $n=1,4,5,6$) were
sent through a turbulent channel and their cross talk quantified using the
average value of the elements along the diagonal of their $4\times 4$
crosstalk matrices. This average, plotted as the fidelity, shows that while
the vacuum modes are perfectly orthogonal in the absence of turbulence, their
fidelity decreases (greater scattering into other modes) for increasing
turbulence strength. However, the eigenmodes of turbulence are not orthogonal
in the absence of turbulence, and their fidelity is consistently worse than
the vacuum modes when passed through a medium for which they were not
designed. This highlights an important aspect of the eigenmodes - they are
robust so long as the medium for which they were created is valid.

##  III Experimental Results

Figure 8: Experimental setup. Lenses L1 and L2 expand and collimate a laser
beam onto an SLM on which a hologram of the initial beam is displayed. The
ideal, turbulence-free beam is generated at this plane and subsequently
propagates through three turbulent screens which are also displayed on SLMs,
each followed by 1 m of free-space propagation. The final aberrated field is
captured on a CCD to image its intensity. The panels on the right show
examples of the desired eigenmodes (Calculated Eigenmodes), the holograms to
create them, and the measured eigenmodes without any turbulence or propagation
(Generated Eigenmodes).

The experiment, shown in Fig. 8, is conceptually divided into three parts. In
the generation stage, a He-Ne laser beam (wavelength $\lambda=633$ nm) was
expanded using a 10$\times$ objective lens L1 and then collimated by L2
($f_{2}$ = 150 mm) before being directed onto a reflective PLUTO-VIS HoloEye
spatial light modulator (SLM) which generated the desired initial field. This
field then entered the turbulent section of the setup where it passed through
three unit cells, each comprising the same random phase screen and a
propagation distance of one metre. The phase screens were generated using the
sub-harmonic random matrix transform method schmidt2010numerical  and
displayed on the SLMs. This perturbed field was then detected and measured on
a camera (CCD). The panels on the right of the experimental setup in Fig. 8
show four examples of the desired (calculated) eigenmodes, the holograms to
create them by complex amplitude modulation, and the experimental validation
that without any turbulence or propagation, that they are created (generated
eigenmodes) with high fidelity (bottom panel).

Our setup differs from conventional laboratory simulations of turbulence in
that we are able to mimic a thick path, from weak to strong turbulence,
whereas often only a single phase screen is used, allowing only weak
turbulence to be tested. Using our setup, we studied an effective real-world
channel of $L=100\text{ m}$, at our wavlength of $\lambda=633\text{ nm}$ and
with Rytov variances of $\sigma^{2}_{R}=1.5,\,1\text{ and }0.5$, corresponding
(respectively) to strong, medium and weak turbulence, with Fried parameters
($r_{0}$) of $0.47\text{ mm, }0.62\text{ mm and }0.93\text{ mm}$,
respectively. We required three screens for each turbulence strength,
separated by a distance of $33.3$ m, each with effective Fried parameters
$r_{0,s}=0.9\text{ mm, }1.2\text{ mm and }1.8\text{ mm}$ while maintaining a
Rytov variance in each slab (segment of the channel) to be smaller than 0.9,
0.6 and 0.3, respectively. This channel was simulated on the setup shown in
Fig. 8 using the Fresnel scaling procedure rodenburg2014simulatingnew ,
allowing a long path to be generated within laboratory dustances. The scaling
factors were chosen to be: $\alpha_{x}=\alpha_{x^{\prime}}=\sqrt{0.03}\approx
0.173$ and $\alpha_{z}=0.03$. This corresponded to a total path length of
$L^{\prime}=3\text{ m}$ and segment Fried parameters of $r_{0,s}=0.081\text{
mm, }0.11\text{ mm and }0.16\text{ mm}$ (see Appendix for details).

The results of OAM and the eigenmodes for weak, medium and strong turbulence
are shown in Fig. 9. The collage shows the final measured eignemodes after the
channel, with the insets showing the initial mode as prepared prior to the
channel. The robustness of the eigenmodes is clearly evident, in contrast to
the highly distorted OAM modes.

Figure 9: Experimental eigenmodes. In the left column we see the measured
intensities of the eigenmodes of turbulence after propagating through the
experimental setup. The results show eigenmodes of weak, medium and strong
turbulence. In the right column we see the measured intensities of OAM modes
after propagating through the same experimental setup with the same turbulence
phase screens for comparison. The insets show the initial input mode. Weak
turbulence was characterised by $\sigma^{2}_{R}=0.5$ and $r_{0}=1.8\text{
mm}$, medium turbulence by $\sigma^{2}_{R}=1$ and $r_{0}=1.2\text{ mm}$ and
strong turbulence by $\sigma^{2}_{R}=1.5$ and $r_{0}=0.91\text{ mm}$

##  IV Discussion

We have developed and implemented a procedure for finding modes which are
unchanged after propagating through a turbulent channel by discretising the
Fresnel integral into an operator which can be represented in a discrete
matrix. These modes are eigenmodes in their truest sense, i.e,. they are fixed
under the action of the channel. This differs from a singular value
decomposition procedure shatokhin2020spatial  which does not return
eigenmodes, but instead requires two different basis sets, one for the input
plane and another for the output plane, with the result that the field
patterns change between the transmitting and receiving planes.

A natural feature of the eigenmodes is that they are channel specific. To be
useful in a real-world setting, the transmission should be faster than the
time frame over which the turbulence changes and the slow time evolving
turbulence (typically 100s of Hz) would have to be monitored to determine the
time evolution of the eigenmodes for a continuous transmission channel. One
can surmise that machine learning would be ideally suited to such a task.

##  V Conclusion

The search for states of structured light that are robust to atmospheric
turbulence is a pressing challenge, promising enhanced channel capacity and
reach in free-space optical links. Here, we have outlined a theoretical
approach to finding the complex forms of structured light which are invariant
under propagation through the atmosphere, the true eigenmodes of turbulence,
and confirmed its validity both numerically and experimentally. These
exotically structured eigenmodes need no corrective procedures, are naturally
devoid of deleterious effects and are valid over any path length in the medium
so long as the medium conditions remain valid. Our approach offers a new
pathway for exploiting structured light in turbulence, and can be easily
extended to other noisy channels, such as underwater and optical fibre.

## Disclosures

The authors declare no conflicts of interest.

## Acknowledgments

Andrew Forbes acknowledges funding from the National Research Foundation (NRF)
and the CSIR-NRF Rental Pool Programme.

## Code, Data, and Materials Availability

Code, data and materials are available on request from the corresponding
author.

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## Appendix

The channel parameters, like path length, are highly restricted in the
laboratory setting. This presents an apparent difficulty to experimentally
verifying the eigenmodes. However, a scaling procedure exists
rodenburg2014simulating  which allows us to verify real-world channels in the
laboratory. This procedure is presented below.

The Fresnel integral for the full (real-world) channel of length $L$ is

$\text{U}_{f}(\mathbf{r},L)=\frac{\exp(ikL)}{i\lambda
L}\int\differential[2]\mathbf{r}^{\prime}\;\text{U}_{i}(\mathbf{r}^{\prime})\exp\left(\frac{i\pi}{\lambda
L}\norm{\mathbf{r}-\mathbf{r}^{\prime}}^{2}\right).$ (10)

We then apply the following scaling parameters:
$\mathbf{r}_{\text{lab}}=\alpha_{x}\mathbf{r},\;\mathbf{r}^{\prime}_{\text{lab}}=\alpha_{r^{\prime}}\mathbf{r}^{\prime}$
and $L^{\prime}=\alpha_{z}L$, where $\mathbf{r}_{\text{lab}}$ and
$\mathbf{r}^{\prime}_{\text{lab}}$ are the coordinates used in the experiment.
The diffraction integral becomes

$\begin{gathered}\text{U}_{f}\left(\frac{\mathbf{r}_{\text{lab}}}{\alpha_{r}}\right)=\frac{\exp\left(ikL^{\prime}/\alpha_{z}\right)}{i\alpha_{r^{\prime}}\lambda
L^{\prime}}\int\differential[2]{\mathbf{r}^{\prime}_{\text{lab}}}\text{U}_{i}\left(\frac{\mathbf{r}^{\prime}_{\text{lab}}}{\alpha_{r^{\prime}}}\right)\times\\\
\times\exp\left(\frac{i\pi\alpha_{z}}{\lambda
L^{\prime}}\norm{\frac{\mathbf{r}_{\text{lab}}}{\alpha_{r}}-\frac{\mathbf{r}^{\prime}_{\text{lab}}}{\alpha_{r^{\prime}}}}^{2}\right).\end{gathered}$
(11)

To keep the diffraction equivalent with these scaled coordinates we require
the Fresnel number

$F=\frac{\pi D_{i}D_{f}}{4\lambda L}$ (12)

to be the same in both the full and scaled-down cases, where $D_{i}$ and
$D_{f}$ are the aperture diameters in the initial and final planes,
respectively. This sets $\alpha_{r}\alpha_{r^{\prime}}=\alpha_{z}$ and the
diffraction integral becomes

$\begin{gathered}\frac{\exp\left(ikL^{\prime}\left(1-\frac{1}{\alpha_{z}}\right)\right)}{\alpha_{r}}\text{U}_{f}\left(\frac{\mathbf{r}_{\text{lab}}}{\alpha_{r}}\right)=\exp\left(-\frac{i\pi
r_{\text{lab}}^{2}}{\lambda
f_{r}}\right)\frac{\exp\left(ikL^{\prime}\right)}{i\lambda
L^{\prime}}\times\\\
\times\int\differential[2]{\mathbf{r}^{\prime}_{\text{lab}}}\;\text{U}_{i}\left(\frac{\mathbf{r^{\prime}}_{\text{lab}}}{\alpha_{r^{\prime}}}\right)\exp\left(-\frac{\pi(r^{\prime}_{\text{lab}})^{2}}{\lambda
f_{r^{\prime}}}\right)\exp\left(\frac{i\pi}{\lambda
L^{\prime}}\norm{\mathbf{r}_{\text{lab}}-\mathbf{r}^{\prime}_{\text{lab}}}\right),\end{gathered}$
(13)

where

$\displaystyle f_{r}$
$\displaystyle=\frac{L^{\prime}}{1-\alpha_{r^{\prime}}/\alpha_{r}},$ (14)
$\displaystyle f_{r^{\prime}}$
$\displaystyle=\frac{L^{\prime}}{1-\alpha_{r}/\alpha_{r^{\prime}}}.$ (15)

Setting $\alpha_{r}=\alpha_{r^{\prime}}$ means that the final and initial
planes have the same size in the laboratory setting and
$f_{r,r^{\prime}}\rightarrow\infty$ and the final Fresnel integral, ignoring
constant phase factors which arise due to scaling, becomes

$\begin{gathered}\text{U}_{f}\left(\frac{\mathbf{r}_{\text{lab}}}{\alpha_{r}}\right)=\frac{\alpha_{r}\exp(ikL^{\prime})}{i\lambda
L^{\prime}}\int\differential[2]{\mathbf{r}^{\prime}_{\text{lab}}}\;\text{U}_{i}\left(\frac{\mathbf{r^{\prime}}_{\text{lab}}}{\alpha_{r^{\prime}}}\right)\\\
\times\exp\left(\frac{i\pi}{\lambda
L^{\prime}}\norm{\mathbf{r}_{\text{lab}}-\mathbf{r}^{\prime}_{\text{lab}}}\right).\end{gathered}$
(16)