Universal Crosstalk of Twisted Light in Random Media

David Bachmann,1,* Asher Klug ,2 Mathieu Isoard,1,† Vyacheslav Shatokhin ,1,3 Giacomo Sorelli ,4,‡

Andreas Buchleitner,1,3 and Andrew Forbes 2

1Physikalisches Institut, Albert-Ludwigs-Universität Freiburg, Hermann-Herder-Straße 3, D-79104 Freiburg, Germany
2School of Physics, University of the Witwatersrand, Private Bag 3, Johannesburg 2050, South Africa
3EUCOR Centre for Quantum Science and Quantum Computing, Albert-Ludwigs-Universität Freiburg,

Hermann-Herder-Straße 3, D-79104 Freiburg, Germany
4Laboratoire Kastler Brossel, Sorbonne Université, ENS-Université PSL, Collège de France,

CNRS; 4 place Jussieu, F-75252 Paris, France

(Received 24 June 2022; accepted 12 December 2023; published 5 February 2024)

Structured light offers wider bandwidths and higher security for communication. However, propagation
through complex random media, such as the Earth’s atmosphere, typically induces intermodal crosstalk.
We show numerically and experimentally that coupling of photonic orbital angular momentum modes is
governed by a universal function of a single parameter: the ratio between the random medium’s and the
beam’s transverse correlation lengths, even in the regime of pronounced intensity fluctuations.

DOI: 10.1103/PhysRevLett.132.063801

Introduction.—Optical communication strives to answer
the growing demand of high-bandwidth links [1–3]. In
particular, photonic spatial degrees of freedom such as
orbital angular momentum (OAM) offer an unbounded
Hilbert space to encode information, and provide intrinsic
support for quantum key or entanglement distribution
protocols [4–7], which with high-dimensional multiplexing
enables greater channel capacities [8] and enhanced secu-
rity [9]. While communication based on such twisted
photons was successfully demonstrated for tabletop [10],
indoor [11], and short outdoor [12] channels, transport
of photonic OAM through complex random media, e.g.,
the atmosphere [13–15], water [16,17], or multimode
fiber [3,18], remains challenging: stochastic fluctuations
of the underlying media’s refractive index induce phase
distortions as well as intensity fluctuations upon propaga-
tion, leading to transmission losses and power transfer
from information encoding modes to others—intermodal
crosstalk—that hinders reliable identification of the input
modes [19,20]. Hence, for a successful optical communi-
cation in complex random media we need to better under-
stand intermodal crosstalk therein.
To that end, we consider OAM-carrying Laguerre-

Gaussian (LG) modes, which are most commonly used
for classical and quantum communication [1,3–7,10–22], as
well as Bessel-Gaussian (BG)modes [23,24]. Information is
typically encoded into the mode’s OAM associated with the
azimuthal index l∈Z [25]; in addition, the transverse
amplitude distribution furnishes another degree of freedom
characterized for LG [26,27] and BG [28] modes, respec-
tively, by the discrete radial indexp∈N0 and the continuous
radial wave number β∈Rþ.
Their rich transverse intensity structures bring about

various applications of twisted photons. For example, LG

modes with p ¼ 0 have the minimal space-bandwidth
product and are usually employed in OAM-multiplexed
systems [29]. Furthermore, the doughnutlike intensity
distribution of such modes is relatively robust under
perturbations [21,22]. On the other hand, LG modes with
p > 0 allow for a substantial increase of the bandwidth
capacity [30]. As for BG modes, they are promising due to
their self-healing [31] and resilience [24,32] properties.
In this Letter, we show that twisted photons, i.e., LG and

BG modes, exhibit a universal dependence of the crosstalk
between modes with opposite OAM governed by the
transverse length scale ratio of the beam and medium.
The model.—We consider transverse optical modes ψðrÞ

arising within the paraxial approximation as solutions of the
parabolic wave equation describing free diffraction [33].
Twisted lightmodes are cylindrically symmetric solutions of
this equation [34]. They feature distinct phase and intensity
profiles that may be characterized by the transverse corre-
lation length ξ [35] and by their l times intertwined helical
phase fronts along the propagation axis z, as illustrated in
Fig. 1(a).
However, photonic phase fronts are fragile under inher-

ent refractive index fluctuations of the atmosphere or of
other complex media. Here, we consider the propagation of
a monochromatic laser beam through clear atmospheric
channels, or through nonabsorbing Gaussian media. In
either case, the typical size of refractive index inhomoge-
neities is much larger than the laser wavelength, so that
wave scattering is mainly in the forward direction [33,36].
We choose the beam’s wavelength to match the infrared
transparency window of atmospheric turbulence [33],
where wave attenuation due to absorption and multiple
scattering by molecules and aerosols over propagation

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distances of a few kilometers is negligible [33,37]. In
other words, our propagation distance L in the atmosphere
is much shorter than the light’s mean free path l [37,38],
and the same regime is assumed for the Gaussian medium.
The beam’s remaining sensitivity to refractive index
fluctuations is modeled by the stochastic parabolic equa-
tion, which incorporates random noise induced by the
medium [33,39]. To find its solution, we employ the split-
step method [40,41], which relies on segmenting the entire
propagation path into discrete, medium-induced phase
modulations, i.e., phase screens [see Fig. 1(a)] intercon-
nected by free diffraction [40,42,43].
The phase screens incorporate the medium’s transverse

correlation length. For atmosphericKolmogorov turbulence
this characteristic length scale is known as the Fried
parameter r0 [33]; the ratio w ≔ w0=r0, with the beam
waist w0, quantifies the resulting distortion strength.
Similarly, the Rytov variance σ2R quantifies intensity fluc-
tuations and distinguishes betweenweak scintillation σ2R <1

and strong scintillation σ2R ≥ 1 [36]. We ensure that every
propagation segment satisfies σ2R < 1. Furthermore, we
represent more general random media by Gaussian noise
based on normally distributed block matrices with block
size r0, independently altering both amplitude and phase of
given transverse modes, where the block size r0 mimics the
correlation length of Kolmogorov turbulence. In this case,
we suppress systematic anisotropies, e.g., due to distinct
lengths of a block’s side and diagonal, by randomly shifting
and rotating the resulting noise grid within the transverse
plane. Examples of a perturbed LG beam are shown in
Fig. 2, where we distinguish three regimes of distortion:
weak, moderate, and strong, quantified by w < 1, w ≈ 1,
and w < 1, respectively.
To benchmark our theoretical description, we experi-

mentally realize a turbulent link by downscaling realistic
channels to tabletop dimensions [44]. The setup, shown in
Fig. 1(b), allows to experimentally simulate propagation
through weak to strong scintillation. The salient elements
are divided into three stages. In the generation stage, a He-
Ne laser beam is expanded and collimated before being

directed onto a reflective spatial light modulator (SLM1)
which generates the desired source mode. This mode then
enters the distorting section of the setup where it passes
through a random medium implemented by a two stage
split-step propagation, with phase shifts programmed to
SLM2 and SLM3, each followed by 1 m of free diffraction.
This 2 m laboratory system maps to a L ¼ 200 m real-
world channel with Rytov variances of up to σ2R ≈ 1.8,
matching the numerically investigated conditions. In
the final stage, the modal decomposition is performed
optically [45] with the aid of a match filter programmed to
SLM4 and a Fourier transforming lens, with the on-axis
intensity measured with a camera (CCD detector).
Crosstalk among distorted modes.—The probability of

identifying a source mode’s original OAM l after propa-
gation as l0 is quantified via the crosstalk matrix

Cl;l0 ðLÞ ≔ jhũlðLÞ; ul0 ðLÞij2; ð1Þ

where ũlðLÞ and ul0 ðLÞ are complex modes [46] propa-
gated across a random medium and through a vacuum
channel each of length L, respectively, and h·; ·i denotes
the standard scalar product in the transverse space at
z ¼ L [47]. The distorted modes ũlðLÞ are connected to
the incident modes ul by the transmission operator TðLÞ,
which approximates the medium’s scattering matrix [48]
whenever wave reflection is neglected. Although geo-
metric truncation due to finite size apertures induces
nonunitarity of TðLÞ, its eigenvalues in weak turbulence
exhibit the bimodal distribution [49] that characterizes
complex scattering media [48]. Accordingly, the crosstalk
matrix can be interpreted as the square modulus of the
elements of the transmission operator’s matrix represen-
tation in the bases of the incident modes and their
vacuum-propagated images.
Consequently, higher transmission fidelities correspond

to weaker crosstalk, i.e., to matrices Cl;l0 ðLÞ with prevail-
ing diagonals. Figure 1(c) sketches the crosstalk amplitudes
for a fixed source mode LGl¼3

p¼0 after propagation through a
channel of length L, for three distortion regimes quantified

FIG. 1. (a) Incident LGl¼3
p¼0 beam with transverse correlation length ξ propagating through a random medium represented by a phase

screen with correlation length r0. (b) Experimental implementation of the split-step method via spatial light modulators (SLMs).
(c) Qualitative illustration of crosstalk Cl;l0 of an incident LGl¼3

p¼0 mode, after propagation through a random medium to z ¼ L, and of

the vacuum-propagated LGl0
p¼0ðz ¼ LÞ modes, for increasing distortion strength w ¼ w0=r0 from top to bottom. Original and opposed

OAM, i.e., �l, are indicated by the solid and dashed black lines, respectively.

PHYSICAL REVIEW LETTERS 132, 063801 (2024)

063801-2



by w. As observed previously [50], under weak distortion
(top row) the detection probability of the original OAM
l ¼ 3 (solid black line) is highest and crosstalk is limited to
neighboring modes. In contrast, under strong distortion [see
bottom row of Fig. 1(c)], we find a �l-symmetric
distribution of the crosstalk amplitudes with separate
maxima close to �l (black lines). In this case, the mode’s
phase front is destroyed, while its amplitude—which is on
average independent of the sign of OAM—partially sur-
vives. Finally, the moderate distortion regime [see center
row of Fig. 1(c)] is of particular interest: it represents the
transition between rather faithful OAM transmission (top

row) and dominant phase destruction (bottom row). Both,
the beam’s amplitude and phase profiles are altered, but
neither is completely destroyed.
To elucidate the origin of the disorder-induced crosstalk

between OAM modes, we decompose the incident LGl
p,

yielding ũlðLÞ in Eq. (1) after transmission through
moderate, w ¼ 1, disorder at z ¼ L, into the vacuum-
propagated LG modes by setting ul0 ðLÞ ¼ LGl0

p ðz ¼ LÞ
in Eq. (1) as illustrated for p ¼ 0 in Fig. 3. Furthermore, we
quantify the channel’s imprint on the transmitted beam’s
phase and intensity profile alone by setting ul0 ðLÞ to
arg½LGl0

p ðz ¼ LÞ� and jLGl0
p ðz ¼ LÞj in Eq. (1), respec-

tively [see left and right columns of Figs. 3(a) and 3(b)].
Note that, in general, randommedia also induce coupling to
modes with p0 ≠ p, but the prescribed projection onto
individual radial indices is common in communication
scenarios [51,52], and required when finite-size apertures
geometrically truncate modes with larger p, or if both, p
and l, are used for information encoding.
The phase profile [see top left plots in Figs. 3(a) and 3(b)],

of the crosstalk matrix concentrates about the diagonal,
which manifests the prevailing coherent coupling of
modes with matching OAM, i.e., l0 ¼ l, confirming pre-
vious findings [50]. Furthermore, the diagonal’s broadening
attests to disorder-induced coupling among modes with
different OAM, i.e., l0 ≠ l. In contrast, when projecting
onto the amplitude profiles [see top right plots in Figs. 3(a)
and 3(b)], crosstalkmatrices exhibit pronounced, symmetric
diagonal and antidiagonal structures. Their symmetry is
expected because the amplitude of LGmodes is independent
of the sign of OAM, and their shape is very suggestivewhen
considering the projection onto LG modes [see top middle

FIG. 3. Numerically obtained average (2500 realizations) crosstalk matrices in moderate (w ¼ 1) Kolmogorov turbulence (a) and
Gaussian noise (b). The crosstalk matrices (top) are obtained when projecting the image ũlðLÞ of the incident mode LGl

p¼0, onto

vacuum-propagated modes ul0 ðLÞ ¼ LGl0
p¼0ðz ¼ LÞ in the middle columns, according to Eq. (1). Left and right columns represent the

beam’s phase and intensity profiles’ modifications quantified by substitution of arg½LGl0
p¼0ðz ¼ LÞ� and jLGl0

p¼0j for ul0 ðLÞ in Eq. (1),
respectively. Rows Cl¼5;l0 (bottom) depict numerical (orange curve, error bands give 1 standard deviation) as well as experimental
results (blue bars, 60 realizations). The crosstalk matrices are normalized such that

P
10
l;l0¼−10 Cl;l0 ¼ 1.

FIG. 2. Intensity-weighted transverse phase profiles of an
incident LGl¼5

p¼0 mode, after propagation through Kolmogorov
turbulence (top), or through Gaussian noise (bottom), for three
different distortion strengths w ¼ 1=2, 1, 2, corresponding to
σ2R ¼ 0.24, 0.76, 2.42 from left to right.

PHYSICAL REVIEW LETTERS 132, 063801 (2024)

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plots in Figs. 3(a) and 3(b). Coherent coupling of neighbor-
ing modes, due to matching OAM phase is combined with
the antidiagonal crosstalk originating from amplitude over-
lap, leading to the presence of damped antidiagonal ele-
ments in agreement with Ref. [52].
The lower row of Fig. 3 compares experimental mea-

surements (blue bars) with numerical results (orange
curves) in form of crosstalk matrix rows that describe
the detection probabilities of various l0 when transmitting
l ¼ 5, cf. orange lines in the upper plots. Experimental
results qualitatively agree with numerical results, and their
systematic underestimation for larger l0 originates from the
receiving aperture damping the overall transmission of
these correspondingly wider modes [33]. In the bottom
middle plots of Figs. 3(a) and 3(b), i.e., when projecting
onto full LG modes, we observe an asymmetric doublet
whose second diminished peak is systematically shifted
toward the center. While crosstalk between �l is still
amplified, this behavior further highlights that we do not
observe direct �l coupling of LG modes but rather a
tradeoff between destruction of OAM-preserving phase and
surviving symmetric amplitude.
We note that antidiagonal, i.e., �l, crosstalk becomes

weaker for larger jlj, cf. fading of antidiagonals in the top
middle plots of Figs. 3(a) and 3(b). This observation is
consistent with well-known results that entangled photonic
OAM qubit states with opposite l become more robust in
turbulence as jlj increases [53]. Careful study of LG
modes [35] within the weak scintillation regime attributed
this enhanced robustness to the finer transverse spatial
structure of such photons characterized by their analytically
given phase correlation length ξ. The latter gives the average
distance between two points in the transverse profile of LG
beams with a phase difference of π=2 [35,54].
Because of its ubiquity in communication protocols

[11,15,35,53,55,56], we further explore the antidiagonal

crosstalk for a range of distortion strengths w and azimuthal
indices l of incident OAM modes. To this end, we inves-
tigate the ratio between antidiagonal and diagonal crosstalk,
i.e., Cl;−l over Cl;l. For LG modes, this ratio is plotted in
Fig. 4, versus the medium’s transverse correlation length r0
normalized by ξ for Kolmogorov turbulence and Gaussian
noise. Further modes are considered in the Supplemental
Material [47]. Remarkably, the rescaling of r0 collapses
the data onto a universal, i.e., l-independent, crosstalk
curve, even in the regime of strong scintillation. Given the
analogy between optical wave transmission and electronic
transport [57], this behavior is reminiscent of the universal
conductance fluctuations observed for electrons in solid
state physics [48],where scattering properties are a universal
function of L=l. However, the latter universality occurs in
the diffusive regime, L=l ≫ 1, which is opposite to the one
here considered.
To ease the comparison and to highlight the universality,

the data points were fitted by a Gaussian [58] with decaying
offset, to mimic the slow systematic noise reduction with
decreasing distortion strength. It is notable that the two
considered, i.e., Kolmogorov and Gaussian, media, albeit
fundamentally different, result in a very similar crosstalk
decay, which is attributed to their similar transverse
length scale.
The crosstalk ratio in Fig. 4 reflects the crossover from

symmetric doublets in strong distortion, where OAM-
encoding phase information is destroyed, to dominant
direct l coupling in weak distortion. Moreover, the
numerical results are in quantitative agreement with exper-
imental data for Kolmogorov turbulence [see circles in
Fig. 4(a)] in the moderate and strong turbulence regime; the
measurement for weak turbulence [see rightmost circle]
shows a systematic offset due to the decreased signal-to-
noise ratio of the diminished antidiagonal crosstalk in this
case. The universality of crosstalk for both Kolmogorov

FIG. 4. Ratio of average (2500 realizations) crosstalk between two OAM-opposed (i.e., l and −l) modes propagated through
numerically simulated Kolmogorov turbulence (a) and Gaussian noise (b) versus the ratio of the medium’s and beam’s transverse
coherence lengths for a range of azimuthal indices jlj. The incident LGl

p¼0 were projected onto LG
∓l
p¼0ðz ¼ LÞ at the receiver side. The

emerging universal curve is fitted with a Gaussian (black curves) with parameters in [58]. Circles with error bars in (a) represent
experimentally measured average (60 realizations) crosstalk in Kolmogorov turbulence. Insets illustrate rows at l ¼ 5 of corresponding
crosstalk matrices (same axes as in Fig. 3). Error bands or bars give 1 standard deviation.

PHYSICAL REVIEW LETTERS 132, 063801 (2024)

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turbulence and generic Gaussian noise suggests that such
agreement can also be expected for other random media.
Discussion.—The universal dependence of crosstalk

amplitudes betweenmodes of oppositeOAMon the rescaled
Fried parameter was previously predicted in the regime
of weak scintillation, for Kolmogorov [35] and non-
Kolmogorov turbulence [56] alike, as well as under
deterministic perturbation of twisted photons by angular
apertures [59]. Moreover, it was established that in this
regime bipartite entanglement of photonic qubit states with
opposite OAM depends on the same parameters as crosstalk
and, hence, exhibits universal behavior as well [35,56,59].
However, entanglement of twisted photons depends on
intensity fluctuations [60] and this sensitivity renders
rescaling into a universal entanglement evolution impossible
under strong scintillation. Our present theoretical and
experimental results show that crosstalk among structured
light modes is nonetheless universal beyond weak scintilla-
tion conditions. Importantly, only the characteristic trans-
verse correlation length of the medium-induced errors
needs be known to understand the impact of modal scatter-
ing on twisted light modes, with clear impact in imaging
and communication through noisy channels. This result
in particular implies that the ubiquitous superposition of
�l [11,15,35,53,55,56], is far from an optimal choice of
encoding information.
Conclusion.—In this Letter, we have studied the crosstalk

of twisted photons in Kolmogorov turbulence and Gaussian
noise. In both cases, we have numerically as well as
experimentally confirmed pronounced crosstalk among LG
modes of opposed OAM. Instead of originating from direct
OAM cross coupling, this behavior was identified as a
tradeoff effect between matching intensity patterns and
destroyed phase information. Moreover, we have uncovered
a universal crosstalk decay complying with experimental
measurements by setting the random media’s transverse
correlation length into relation with the beam’s phase struc-
ture. We have established that the universality holds for LG
modes with different radial indices, that is, with different
transverse amplitude distributions, as well as for BG modes
with different beam waists [47]. We therefore envisage a
similar behavior for other sets ofOAMmodes upon a suitable
generalization of the phase correlation length [35,47].
Our results may lead to novel venues for communication.

In particular, optimizing the beam’s transverse correlation
length for given distortion strengths will diminish the
crosstalk, irrespective of the precise nature of the under-
lying random medium itself. As we have demonstrated, this
can be achieved by suitable choice of l or p (β) for LG
(BG), or, due to the scaling properties of LG and BG
modes, by adapting the beam waist w0.

D. B. acknowledges financial support by the Studien-
stiftung des deutschen Volkes. The authors acknowledge
support by the state of Baden-Württemberg through bwHPC

and the German Research Foundation (DFG) through Grant
No. INST 40/575-1 FUGG (JUSTUS 2 cluster). G. S.
acknowledges partial funding by French ANR (ANR-19-
ASTR0020-01). V. S. and A. B. acknowledge partial
funding and support through the Strategiefonds der
Albert-Ludwigs-Universität Freiburg and the Georg
H. Endress Stiftung.

*Corresponding author: david.bachmann@physik.uni-
freiburg.de

†Present address: Laboratoire Kastler Brossel, Sorbonne
Université, ENS-Université PSL, Collège de France,
CNRS, 4 place Jussieu, F-75252 Paris, France.

‡Present address: Fraunhofer IOSB, Ettlingen, Fraunhofer
Institute of Optronics, System Technologies and Image
Exploitation, Gutleuthausstraße 1, D-76275 Ettlingen,
Germany.

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