# High-order aberrations of vortex constellations

R. F. Barros  Tampere University, Photonics Laboratory, Physics Unit, Tampere,
FI-33720, Finland<EMAIL_ADDRESS>S. Bej  Tampere University, Photonics
Laboratory, Physics Unit, Tampere, FI-33720, Finland  M. Hiekkamäki  Tampere
University, Photonics Laboratory, Physics Unit, Tampere, FI-33720, Finland  M.
Ornigotti  Tampere University, Photonics Laboratory, Physics Unit, Tampere,
FI-33720, Finland  R. Fickler  Tampere University, Photonics Laboratory,
Physics Unit, Tampere, FI-33720, Finland

###### Abstract

When reflected from an interface, a laser beam generally drifts and tilts away
from the path predicted by ray optics, an intriguing consequence of its finite
transverse extent. Such beam shifts manifest more dramatically for structured
light fields, and in particular for optical vortices. Upon reflection, a field
containing a high-order optical vortex is expected to experience not only
geometrical shifts, but an additional splitting of its high-order vortex into
a constellation of unit-charge vortices, a phenomenon known as topological
aberration. In this article, we report on the first direct observation of the
topological aberration effect, measured through the transformation of a vortex
constellation upon reflection. We develop a general theoretical framework to
study topological aberrations in terms of the elementary symmetric polynomials
of the coordinates of a vortex constellation, a mathematical abstraction which
we prove to be the physical quantity of interest. Using this approach, we are
able to verify experimentally the aberration of constellations of up to three
vortices reflected from a thin metallic film. Our work not only deepens the
understanding of the reflection of naturally occurring structured light fields
such as vortex constellations but also sets forth a potential method for
studying the interaction of twisted light fields with matter.

††preprint: APS/123-QED

Introduction.— The reflection of light beams on interfaces is a quintessential
problem in wave optics. In contrast to simple geometrical optics laws, where a
ray of light reflects of a surface with the same incidence angle but on the
opposing side of the surface normal, the reflection of beams in wave optics
encompasses a more complex behavior. Generally, a beam can be mathematically
described by a finite spectrum of plane waves. Upon reflection, each of these
plane waves individually follows paths determined by geometrical optics [1].
However, the changes to the amplitudes and phases of the component plane waves
result in a reflected field which is macroscopically different from the
incident field. Perhaps the most well-known examples of such phenomena are
optical beam shifts, which are spatial and angular deviations of reflected
laser beams from their expected ray-optical trajectories [2, 3]. Such shifts
are usually separated into Goos-Hänchen (GH) [4, 5] and Imbert-Fedorov (IF)
[2, 6] shifts, which occur along and orthogonal to the plane of incidence,
respectively. These effects have distinct physical origins: while the GH
shifts come from the angular variation of the reflection coefficient of the
interface, the IF shifts, also known as the spin-Hall effect of light [7],
stem from the rotation of the plane of incidence experienced by elliptically
polarized waves upon reflection. The GH and IF shifts have been extensively
studied for different types of optical beams and interfaces in a range of
contexts, as discussed in detail in Ref.[2].

A more subtle topic is the reflection of vortex beams [2, 8, 9, 10, 11, 12,
13]. Such beams are most well known for their ability to carry orbital angular
momentum (OAM), which is accompanied by a twisted phase structure, i.e., a
varying phase front transverse to the propagation direction as
$\exp(i\ell\phi)$, where $\ell$ is an integer number and $\phi$ labels the
azimuthal angle [14, 15]. The wavefronts of such beams consist of $|\ell|$
identical helicoids nested on the propagation axis, on which lies a
strength-$\ell$ optical vortex or phase singularity [16, 17]. Due to the OAM-
induced mixing of the spatial and angular GH and IF shifts [18], vortex beams
also acquire OAM sidebands upon reflection [19] and feature significant
deformations in their intensity profiles [8, 9] at critical angles. Beyond
beam shifts, Dennis and Götte showed in their seminal work [20] that a pure
strength-$\ell$ optical vortex splits into a constellation of unit strength
vortices by reflecting at a simple dielectric interface, which they recognized
as a topological aberration effect. It is well known, however, that it is
impossible to generate perfect higher-order vortices, as the latter are
unstable under any kind of pertubation[21, 22, 23]. Therefore, a fundamental
question remains open: how do vortex constellations, the actual observable
physical objects, experience topological aberrations?

In this article, we adress this question first by generalizing the framework
developed in Ref. [20] to arbitrary uniform aberrations of vortex
constellations, thereby addressing typical experimental limitations in the
generation of vortex beams. We then show that the aberration of a vortex
constellation is captured by a linear transformation of the elementary
symmetric polynomials of its coordinates, and that this transformation is
related to the angular Wirtinger derivatives of the aberration. Lastly, we
detail the experimental observation of the topological aberration of vortex
constellations under total internal reflection from a thin Au film-Fused
Silica interface, where aberration effects are enhanced due to the resonant
excitation of surface plasmons at the interface. With this method, we are able
to verify the topological aberration using constellations with up to $3$
vortices, in good agreement with the theoretical model developed. Due to the
direct link between the vortex dynamics and the properties of the material
upon which the light is reflected, our results could be applied to advanced
material characterization techniques. Moreover, the underlying theoretical
description of the vortex dynamics might also be applicable to other fields of
physics such as Bose-Einstein condensates [24, 25], superfuilds[26, 27], or
topological field theories[28].

Theory of topological aberrations.— We start by considering an arbitrary input
field containing a constellation of $\ell_{m}$ identical unit-strength
vortices. In momentum space, the scalar part of such a field can be written as
[20]

$\tilde{\psi}_{I}(\bm{\chi})=\sum_{\ell=0}^{\ell_{m}}\sigma_{\ell}(|\chi|)\,\left(\frac{\chi}{|\chi|}\right)^{\ell}\,,$
(1)

where $\bm{\chi}=(\chi,\chi^{*})$, with $\chi=(k_{x}+ik_{y})/\sqrt{2}$, and
$(k_{x},k_{y})$ are Cartesian coordinates in momentum space.
$\sqrt{2}|\chi|=k_{\perp}=\sqrt{k_{x}^{2}+k_{y}^{2}}$ and
$\textrm{Arg}(\chi)=\alpha=\tan^{-1}(k_{y}/k_{x})$, where $k_{\perp}$ and
$\alpha$ are the momentum radial and azimuthal coordinates, respectively.
Equation (1) can be seen as a superposition of optical vortices with
topological charges $0\leq\ell\leq\ell_{m}$ , whose background functions
$\sigma_{\ell}(|\chi|)$ determine both the OAM spectrum and the field’s radial
features. Furthermore, the constellation coordinates are encoded in the
complex roots of the field (1), which is a polynomial of order $\ell_{m}$ in
$\chi$.

A field of the form given in Eq. (1), i.e. a constellation of vortices, is
naturally obtained by attempting to generate a vortex of order $\ell_{m}$
experimentally, due to inherent limitations in light-shaping devices and/or
subsequent aberrations caused by mirrors, lenses, and other refractive
elements. Nevertheless, for small aberrations, which is the case for a
carefully assembled experiment, the contributions of $\ell<\ell_{m}$ are small
and the constellation is tightly confined to the beam’s central propagation
direction. In this case, we can approximate the field in Eq. (1) in real space
to the lowest order of each OAM component, obtaining

$\psi_{I}(\bm{\xi})=\sum_{\ell=0}^{\ell_{m}}\bar{\sigma}_{\ell}\xi^{\ell}\,,$
(2)

where $\bm{\xi}=(\xi,\xi^{*})$, with $\xi=(x+iy)/\sqrt{2}$, and where $(x,y)$
are the Cartesian coordinates in real space. The coefficients
$\bar{\sigma}_{\ell}$, derived in detail in the Supplementary Material,
represent the $\ell$-th order moments of the background functions
$\sigma_{\ell}(|\chi|)$.

Upon a spatially uniform aberration such as the reflection from a flat
interface, each plane wave component of momentum $\bm{\chi}$ in the angular
spectrum (1) is transformed as
$\exp(i\bm{\chi}\cdot\bm{\xi}^{*})\rightarrow\,R(\chi)\exp(i\chi\cdot\xi^{*})$,
where $R(\bm{\chi})$ is the momentum-dependent aberration function. In the
case of reflection from a flat interface, the aberration function is the
reflection coefficient given by the Fresnel coefficients, which also depends
on the incident and measurement polarizations [20, 2]. The resulting field in
real space can then be modeled as

$\displaystyle\psi_{R}(\bm{\xi})=\int
d^{2}\bm{\chi}\tilde{\psi}_{I}(\bm{\chi})R(\bm{\chi})\exp(i\bm{\chi}\cdot\bm{\xi}^{*})\,,$
(3)

which is a Fourier transform in the complex coordinates $\bm{\xi}$ and
$\bm{\chi}$. Hence, we can expand the aberration in a power series and use the
differentiation property of the Fourier transform to obtain

$\displaystyle\psi_{R}(\bm{\xi})$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\sum_{m=0}^{n}i^{-n}C^{m}_{n}\frac{\partial_{n}\psi_{I}(\bm{\xi})}{\partial\xi^{*m}\partial\xi^{n-m}}\,,$
(4) $\displaystyle C^{m}_{n}$ $\displaystyle=$
$\displaystyle\frac{1}{n!}\binom{n}{m}\frac{\partial_{n}R(\bm{\chi})}{\partial\chi^{m}\partial\chi^{*n-m}}\Bigg{|}_{\chi,\chi^{*}=0}\,,$
(5)

where ${\partial}/{\partial\xi}=\left({\partial}/{\partial
x}-i{\partial}/{\partial y}\right)/\sqrt{2}$ and
${\partial}/{\partial\chi}=\left({\partial}/{\partial
k_{x}}-i{\partial}/{\partial k_{y}}\right)/\sqrt{2}$ are Wirtinger derivatives
[29]. From Eqs. (2) and (4), we arrive at the final expression for the
aberrated field

$\displaystyle\psi_{R}(\bm{\xi})$
$\displaystyle=\sum_{\ell=0}^{\ell_{m}}\bar{\gamma}_{\ell}\xi^{\ell}=\sum_{\ell=0}^{\ell_{m}}\sum_{n=0}^{\ell}i^{-n}\bar{\sigma}_{\ell}n!\binom{\ell}{n}C^{0}_{n}\xi^{\ell-n}\,,$
(6)

where $\bar{\gamma}_{\ell}$ are the coefficients of the vortex expansion of
the aberrated field.

Equation (6) shows that an aberration decomposes the incoming light field into
a superposition of its Wirtinger derivatives, weighted by the Wirtinger
derivatives of the aberration function. Interestingly, in the case of the
vortex input field of Eq.(1) the derivative modes are simply optical vortices
of lower order, in such a way that the aberrated field still contains a
collection of $\ell_{m}$ phase singularities, but in a deformed constellation,
as we illustrate in Fig. 1. This implies that by monitoring the changes in a
vortex constellation, one can gain insight into the properties of the
aberration function, and thus into features of the light-matter interaction
causing the aberration.

Figure 1: Conceptual picture of the topological aberration of a vortex
constellation upon reflection. $\vec{k}$ denotes a plane wave component of the
incident vortex, while $\vec{k}_{0}$ is the central wave vector in the beam.
The plane of incidence is $k_{x}k_{z}$, and the dashed lines show the ray
optics trajectory of the central plane wave $\vec{k}_{0}$, with the angle of
incidence $\theta_{0}$. The input and reflected constellations are marked with
black crosses and orange circles, respectively.

High-order aberrations of vortex constellations— Equation (6) establishes a
direct correspondence between the coefficients of the vortex expansions of the
incident and aberrated fields. However, the question still remains as to how
the coordinates of the corresponding constellations compare. This problem is
elegantly solved by Vieta’s theorem [30], which we illustrate in the
following. Consider an arbitrary constellation of $\ell_{m}$ vortices with
coordinates $(x_{j},y_{j})$, which yields a vortex expansion with roots
$\Delta_{j}=(x_{j}+iy_{j})/\sqrt{2}$ and coefficients $c_{i}$ to be
determined. According to Vieta’s theorem we have that

$e_{j}=(-1)^{j}\frac{{c}_{\ell_{m}-j}}{{c}_{\ell_{m}}}\,,\quad
1<j\leq\ell_{m}\,,$ (7)

where $e_{j}$ is the $j$-th order elementary symmetric polynomial (ESP) of the
set of complex roots $\\{\Delta\\}$. In terms of ESPs, equation (6) assumes
the simple form

$\bm{e}_{R}=\hat{R}_{\chi}\bm{e}_{I}\,,$ (8)

where the vectors $\bm{e}_{I}$ and $\bm{e}_{R}$ contain the ESPs of the input
and aberrated constellations, respectively, with the aberration operator
$\hat{R}_{\chi}$ acting on the $\ell_{m}$-dimensional subspace spanned by the
ESPs. The explicit form for the operator $\hat{R}_{\chi}$ is an upper
triangular matrix containing the Wirtinger derivatives of the aberration
function $R(\bm{\chi})$ up to the order $\ell_{m}$, as we detail in the
Supplementary Material.

We conclude from Eq. (8) that the aberration of a vortex constellation is
fully captured by a linear transformation of its ESPs. Furthermore, an
intuitive physical/geometrical interpretation of the ESP transformations is
possible by means of Newton’s identities [30], which relate the ESPs to power
sums of a set of complex roots. For example, $e_{1}$ is proportional to the
constellation barycenter, whose transformation contains the GH and IF beam
shifts of vortex beams [2, 3, 20, 18]. On the other hand, $e_{2}$ and $e_{3}$
give the second and third moments of the constellation positions with respect
to the barycenter, which are related to the variance and the skewness of the
constellation, respectively. For higher orders, the ESPs are sums of products
of moments of the constellation positions that add up to that order, whose
meaning we could not identify.

For clarity, let us explicitly consider the case of $\ell_{m}=2$. We choose
the origin of the reference frame at the barycenter of the input
constellation, such that the input field is
$\psi_{I}(\bm{\xi})\propto\xi^{2}+e_{I2}\xi^{0}$. From Eqs. (6) and (7), we
obtain that

$\displaystyle e_{R1}$ $\displaystyle=$ $\displaystyle
2iR_{\chi}^{\prime}/R_{\chi}\Big{|}_{0}\,,$ (9) $\displaystyle e_{R2}$
$\displaystyle=$ $\displaystyle
e_{I2}-R_{\chi}^{\prime\prime}/R_{\chi}\Big{|}_{0}\,,$ (10)

where $R_{\chi}^{\prime}|_{0}$ and $R_{\chi}^{\prime\prime}|_{0}$ are the
first and second Wirtinger derivatives of $R(\bm{\chi})$ at $\chi=\chi^{*}=0$.
Equation (9) shows the shift of the barycenter of the constellation by the
known Artmann translator of a unit vortex [18, 12], which is the first order
topological aberration of the constellation. In fact, this result, previously
predicted only for perfect input vortices [20], holds regardless of the
particular geometry or the order of a vortex constellation. Furthermore,
equation (10), shows the second-order topological aberration, which amounts to
the stretching of the input constellation and its rotation around the
barycenter.

Figure 2: Experimental setup containing: BP, bandpass filter; ND, neutral
density filter; M1, retroreflecting mirror; PBS, polarizing beam-splitter; BS,
50:50 non-polarizing beam-splitter; HWP1 and HWP2, true-zero order half-wave
plates; SLM, spatial light modulator; CMOS, camera sensor. The inset shows a
breakdown of our reflecting object, which consists of an Au film assembled on
a FS substrate and optically contacted to a FS prism.

Experiment.— To observe the aberration of vortex constellations, we use the
experimental setup depicted in Fig.2. We use a fiber-coupled diode laser
centered at the wavelength $\lambda=810\,$nm, operated below the lasing
threshold and filtered by a narrow bandpass filter (Semrock, 3nm bandwidth).
The beam is then polarized on a polarizing beam-splitter (PBS) and divided
with a non-polarizing 50:50 beam-splitter (BS) into a probe beam, which we
prepare with the desired constellation, and a reference beam, used afterwards
for the generation of off-axis holograms. To prepare the probe beam, we shape
it on a spatial light modulator (SLM, Holoeye Pluto 2.0) displaying a
$\exp(i\ell_{m}\phi)$ phase structure, which naturally leads to a
constellation tightly confined to the beam center. We further tune the
constellation by introducing wavefront aberrations by means of Zernike
polynomials [31], ensuring that all the $\ell_{m}$ vortices are clearly
observable in our apparatus. As an additional precaution, we use a second PBS
to prevent any undesired changes in polarization due to the SLM.

Our aberrating device consists of a 35nm thick Au film assembled on the
hypothenuse of a fused silica (FS) right-angle prism, in the so-called
Kretschmann-Raether (KR) configuration [32, 33]. The Au film is deposited on
an FS substrate using electron beam-assisted evaporation and optically
contacted to the prism with index-matching gel (Thorlabs G608N3). In the KR
configuration, a P-polarized input beam can resonantly excite surface plasmons
at the interface between FS and the Au film via attenuated total reflection
(ATR), enhancing the angular gradients of the reflection coefficient, and
hence the topological aberration imprinted on the reflected beam [34]. By
measuring the power of the reflected S- and P-polarized probe beams as a
function of the angle of incidence and using a transfer matrix model (see
Supplementary Material) to numerically fit the results, we determine the
resonant ATR angle and the effective permittivity of the Au film to be
$\theta_{ATR}\approx 45.02^{\circ}$ and $\epsilon_{f}=-22.8456+1.2619i$,
respectively. Furthermore, since only P-polarized light excites surface waves,
we can use the reflected S-polarized field as a good approximation of the
input field and consider the aberration function $R(\bm{\chi})$ to be the
fitted reflection coefficient for P-polarized light.

Figure 3: a) First, b) second, and c) third Wirtinger derivatives of the
reflection coefficient for P-polarized light, retrieved from the measured
vortex constellations for $1\leq\ell_{m}\leq 3$. In each plot, the real
(imaginary) part is shown as filled (hollow) markers for measured data, and as
solid (dashed) curves for theoretical simulations. Additionally, in d) and e)
we show the measured vortex constellations of the input field (crosses) and
the reflected field (circles) with $\ell_{m}=2$ in d) and $\ell_{m}=3$ in e).
The measured constellations are shown for the 15 angles of incidence
highlighted on the $x$ axis of b) and c).

To measure the constellation coordinates, we superpose the probe and reference
beams off-axis and record the interference patterns with a CMOS camera (IDS
xxx, 2.2 $\mu$m pixel size), from which both the intensity and phase profiles
are digitally retrieved [35, 36]. It is worth noting that the measured
constellations are affected by interference with stray light coming from the
protective glass window on the CMOS sensor, the reflective layers of the beam-
splitters, etc., which cannot be eliminated in post-processing due to the
propagation direction being almost identical to the probe field. The laser was
operated below the threshold in order to eliminate these unwanted interference
effects, at the cost of requiring precise control of the path length
difference between reference and probe beams. In our setup we match the path
lengths with a translation stage, also ensuring negligible lensing effect on
the recorded holograms. Furthermore, we dim the intensity of the reference
beam with neutral density filters, improving the hologram visibility around
the singularity positions and allowing better use of the CMOS dynamic range.
Finally, we determine the constellation coordinates from the retrieved phase
profiles using the algorithm detailed in the Supplementary Material. The
uncertainty of the method was estimated to be $1/2$ pixel irrespective of the
hologram fringe density, which is the uncertainty we used to calculate the
error bars shown in Fig.3.

Results.— We show in Fig 3 our experimental results for constellations with
$1\leq\ell_{m}\leq 3$. In Fig.3a we show the logarithmic Wirtinger derivative
of the reflection coefficient, retrieved from the shifts in the barycenters of
the constellations using Eqs. (6) and (9). We see that the retrieved
derivatives accurately follow the theoretical curves simulated from the
measured material parameters under the assumption $R_{\chi}^{(n)}|_{0}\approx
R_{k_{x}}^{(n)}|_{0}/2^{n/2}$. Such an approximation is reasonable in our
case, since $k_{x}k_{z}$ is the plane of incidence, and the constellations are
prepared and measured in the same polarization. We note that, unlike the
shifts in the center of mass for the vortex-carrying fields [20, 12], the
measured barycenter shifts do not depend on the total topological charge
$\ell_{m}$. This result is a direct measurement of the first-order topological
aberration of a vortex constellation, which, despite its strong correspondence
to the GH and IF shifts, had not yet been observed. Furthermore, the
barycenter shifts we report are greatly enhanced compared to those of
dielectric interfaces, reaching more than $30$ wavelengths in magnitude. A
similar enhancement of the GH shift has been reported in [34] for Gaussian
beams, which here we extend to vortex beams and out-of-plane (IF) shifts.

Coming to the second-order topological aberration, in Fig. 3b we show the
second Wirtinger derivative of the reflection coefficient, retrieved from the
transformations in the second-order ESPs of the constellations with
$\ell_{m}=2$ and $\ell_{m}=3$. The retrieved derivatives agree well with the
theoretical simulations, although the experimental results show sharper
features when compared to the theory. The geometrical meaning of the second-
order topological aberration can also be seen in Fig. 3d, where we show the
measured input and aberrated constellations for the $15$ angles of incidence
highlighted by an arrow in Fig. 3b. We see that near the resonant ATR angle,
both the size and shape of the constellation change noticeably, in addition to
the barycenter shifts following Fig. 3a. Furthermore, we note that the
reflected S-polarized constellations do not change noticeably with the angle
of incidence, supporting our assumption that they are good approximations of
the input constellations.

Lastly, in Fig. 3c we show the third Wirtinger derivative of the reflection
coefficient, retrieved from the changes in the third-order ESPs of the
$\ell_{m}=3$ constellation. In this case, the experimental results diverge
more substantially from the theoretical simulations, which we attribute to two
possible sources of error. First, measuring constellations is increasingly
challenging as the constellation order increases, since the intensity near the
singularity positions decreases exponentially with $\ell_{m}$. For
$\ell_{m}>2$, in our case, properly exposing the hologram near the singularity
positions requires long exposure times, and hence the measurements are more
susceptible to mechanical instabilities and noise. Second, the transfer matrix
model that we used for the theoretical simulations does not include the
roughness of the Au film surface. Due to these limitations, we did not measure
the aberrations of constellations of even higher order. Nonetheless,
analogously to before, we show in Fig. 3e the raw constellation coordinates
for the set of angles highlighted by an arrow in Fig. 3c. Here, the
deformation of the P-polarized constellations as well as the shift of the
barycenter can be clearly seen.

Conclusion.— In this article, we developed a new framework to describe the
topological aberration of vortex constellations and reported on the first
direct observation of the phenomenon predicted more than a decade ago [20]. We
showed that aberrations generally affect the elementary symmetric polynomials
of a constellation’s coordinates, from which the angular Wirtinger derivatives
of the aberration can be directly retrieved. We demonstrate the effect
experimentally by measuring the topological aberrations of vortex
constellations with up to $3$ vortices upon attenuated total reflection, where
aberration effects are enhanced by the resonant excitation of surface plasmon
polaritons on the interface between a thin metallic film and a dielectric
medium. Our results mark the first experimental observation of the topological
aberration effect and introduce a new framework for probing light-matter
interactions with twisted light.

These topological aberration effects, when described with a precise
theoretical model, will be a useful tool to derive the optical properties of
metal-dielectric interfaces, thereby possibly simplifying current standard
characterization techniques such as ellipsometry. In addition, because of the
generality of our underlying theoretical framework describing the dynamics of
vortex constellations, it will be interesting to apply our approach to other
singular light fields featuring, for example, polarization singularities [17],
vortex knots [37] and polarization knots [38]. Beyond light waves, we expect
our work to inspire connections to other fields of physics where complex
vortex constellations appear, such Bose-Einstein condensates, superfluids [24,
25, 26, 27], and even topological field theories [28]. Finally, as the
structured light fields are seen as promising approaches to encode classical
and quantum information, a better understanding / simplified description of
the vortex dynamics will facilitate the reduction of errors caused by
aberrations [39].

Acknowledgements.—We thank Jörg Gotte and Mark Dennis for inspiring
discussions at the ICOAM 2022. We also thank Matias Eriksson for the helpful
comments on the data processing. R. B. acknowledges the support of the Academy
of Finland through the postdoctoral researcher funding (decision 349120). M.
H. acknowledges support from the Doctoral School of Tampere University, Emil
Aaltonen foundation, and the Magnus Ehrnrooth foundation through its graduate
student scholarship R. F. acknowledges the support of the Academy of Finland
through the Academy Research Fellowship (decision 332399). All authors
acknowledge the support of the Academy of Finland through the Competitive
Funding to Strengthen University Research Profiles (decision 301820) and the
support of the Photonics Research and Innovation Flagship (PREIN - decision
320165).

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