# Diffraction imaging of light induced dynamics in xenon-doped helium
nanodroplets

B Langbehn1, Y Ovcharenko1,2, A Clark3, M Coreno4, R Cucini5, A Demidovich6, M
Drabbels3, P Finetti6, M Di Fraia6,4, L Giannessi6, C Grazioli5, D
Iablonskyi7, A C LaForge8, T Nishiyama9, V Oliver Álvarez de Lara3, C Peltz10,
P Piseri11, O Plekan6, K Sander10, K Ueda7, T Fennel10,12, K C Prince6,13, F
Stienkemeier8, C Callegari6,4, T Möller1 and D Rupp1,12,14 1 Technische
Universität Berlin, Institut für Optik und Atomare Physik, 10623 Berlin,
Germany  2 European XFEL GmbH, 22607 Hamburg, Germany  3 Laboratory of
Molecular Nanodynamics, Ecole Polytechnique Fédérale de Lausanne (EPFL), 1015
Lausanne, Switzerland  4 ISM-CNR, Istituto di Struttura della Materia, LD2
Unit, 34149 Trieste, Italy  5 IOM-CNR, Istituto Officina dei Materiali, 34149
Trieste, Italy  6 Elettra–Sincrotrone Trieste, S.C.p.A., 34149 Trieste, Italy
7 Institute of Multidisciplinary Research for Advanced Materials, Tohoku
University, Sendai 980-8577, Japan  8 Physikalisches Institut, Universität
Freiburg, 79104 Freiburg, Germany  9 Division of Physics and Astronomy,
Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan  10
Institut für Physik, Universität Rostock, 18051 Rostock, Germany  11 CIMAINA
and Dipartimento di Fisica, Università degli Studi di Milano, 20133 Milano,
Italy  12 Max-Born-Institut für Nichtlineare Optik und Kurzzeitspektroskopie,
12489 Berlin, Germany  13 Department of Chemistry and Biotechnology, Swinburne
University of Technology, Victoria 3122, Australia  14 Laboratorium für
Festkörperphysik, ETH Zürich, 8093 Zürich, Switzerland
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We have explored the light induced dynamics in superfluid helium nanodroplets
with wide-angle scattering in a pump-probe measurement scheme. The droplets
are doped with xenon atoms to facilitate the ignition of a nanoplasma through
irradiation with near-infrared laser pulses. After a variable time delay of up
to 800 ps, we image the subsequent dynamics using intense extreme ultraviolet
pulses from the FERMI free-electron laser. The recorded scattering images
exhibit complex intensity fluctuations that are categorized based on their
characteristic features. Systematic simulations of wide-angle diffraction
patterns are performed, which can qualitatively explain the observed features
by employing model shapes with both randomly distributed as well as
structured, symmetric distortions. This points to a connection between the
dynamics and the positions of the dopants in the droplets. In particular, the
structured fluctuations might be governed by an underlying array of quantized
vortices in the superfluid droplet as has been observed in previous small-
angle diffraction experiments. Our results provide a basis for further
investigations of dopant-droplet interactions and associated heating
mechanisms.

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Keywords: coherent diffraction imaging, helium nanodroplets, superfluid,
dynamics

##  1 Introduction

The interaction of intense laser pulses with matter plays an important role in
many fields, from modern high-tech applications such as laser machining [1]
and microsurgery [2], extreme ultraviolet (XUV) lithography [3], or propulsion
of small satellites [4, 5] to fundamental and applied science. For example, in
structure analysis of biological specimen using x-ray diffraction, radiation
damage poses a substantial problem, thus fueling a considerable research
interest in the processes accompanying and following the irradiation with
intense x-ray bursts [6].

In particular, atomic clusters can serve as isolated model systems of simple
geometrical structure. Time-resolved studies, where an initial _pump_ light
pulse excites the system and a subsequent _probe_ light pulse measures the
state of the system after a defined temporal delay, enable observation of the
light induced dynamics. In the extreme case, the free cluster will eventually
fully disintegrate, as the energy deposited by the pump pulse cannot dissipate
otherwise. A detailed investigation of the fragments and the dynamics can give
an insight into the processes and their inherent time scale, which, in
comparison with theoretical models of the interaction, enable development of a
fundamental understanding [7, 8, 9, 10, 11, 12]. A basic picture divides the
interaction of an intense light pulse with an atomic cluster into three steps
[13, 14]: (i) the ionization of cluster atoms and emission of electrons until
the increasing positive charge prevents further electrons from leaving the
cluster compound, (ii) the formation and evolution of a nanoplasma of ions and
quasi-free electrons on a short timescale, often only a few femtoseconds, and
(iii) an expansion leading to complete destruction of the cluster, accompanied
by relaxation and recombination processes.

Helium clusters, often referred to as helium nanodroplets, are model systems
particularly suited to such studies due to their very simple electronic
structure and smooth behavior: They remain liquid down to absolute zero
forming mostly spherical shapes [15, 16], and they are transparent from the
far-infrared to the vacuum ultraviolet because of the large ionization
potential of helium ($24.6~{}\mathrm{eV}$). Using a near infrared (NIR) laser
pulse, helium nanodroplets can be ionized only when the power density is
sufficiently high that multiphoton ionization or field driven processes like
tunneling ionization become significant ($\sim
10^{15}~{}\mathrm{W}\mathrm{cm}^{-2}$) [17, 18]. Another option is to dope the
droplets using atoms with a low ionization potential: In this case, ionization
becomes possible already at lower laser intensity [19, 20], starting an
avalanche-like process at the dopant positions [21, 22]. Depending on the
interaction of the impurity with the helium solvent, the dopants remain at the
surface of the droplet or get immersed, as is the case for, e.g., xenon atoms
[23]. When xenon-doped droplets are irradiated by an NIR pulse, theory
predicts a very efficient complete ionization even with only a few xenon atoms
embedded [19]. For the simple model of a compact dopant cluster in the center
of the helium droplet, an anisotropic growth of the nanoplasma along the laser
polarization axis has been predicted upon ignition [19]. On the other hand,
dopant cluster formation has been observed at _multiple_ centers in larger
droplets [24]. Further, the superfluid nature of the helium nanodroplets leads
to the occurrence of quantized vortices in large droplets [15], and the
dopants agglomerate along the vortex lines [25, 26]. Nanoplasma ignition can
therefore be expected in large droplets at multiple sites, randomly
distributed or structured by the vortex positions, which might lead to complex
dynamics. In addition to nanoplasma dynamics governed by Coulombic forces
between charged constituents, geometric changes beyond homogeneous Coulombic
explosion or hydrodynamic expansion might have to be considered in helium
nanodroplets. For example, the formation of voids around free electrons, so-
called _electron bubbles_ , has been described in liquid helium [27]. Further,
the formation of gas bubbles has been observed on a macroscopic scale around
copper nanoparticles in bulk liquid helium [28] and has also been reported on
a microscopic scale around silver clusters in helium nanodroplets [29], in
both cases induced by resonant heating of the metal particles with an optical
laser. Although superfluid helium is known for its very large heat
conductivity, the flow of thermal energy from a solid particle to liquid
helium is hindered by a discrepancy in velocity of sound, an effect called
Kapitza resistance [29, 30], that leads to a sudden evaporation of the
surrounding helium, eventually forming a gas bubble. A third process that
could be taken into account is the ejection of dopant ions from a helium
nanodroplet, leaving the droplet with several helium atoms attached to them
[31, 32]. These considerations show that the study of light induced dynamics
in doped helium droplets may have particularly interesting aspects. So far,
most theoretical work has been carried out for relatively small helium
droplets ($N\leq 10^{6}$ atoms per droplet). While imaging experiments
inherently address larger droplets [15, 16], their outcome has already
stimulated theoretical studies, e.g., on the droplet shapes [33, 34]. It has
been the close interplay between theory and experiment that established a more
profound knowledge of the underlying physics of rotating superfluid droplets
[35]. Therefore, it would be particularly interesting to experimentally
visualize the droplet dynamics after laser excitation, giving a basis for
comparison with theoretical models. Eventually, if droplet ignition at
multiple sites or geometric changes in the droplets are observed, such
experiments might lead to an extension of the models or even to a new approach
to gain further insight into the processes, e.g., by employing molecular
dynamics simulations.

In this work, we report on a study of the dynamics triggered by intense NIR
laser pulses in large, xenon-doped helium nanodroplets. In this context, the
advent of short-wavelength free-electron lasers (FELs) has opened up a new
route to determine the structure of individual nanoparticles via coherent
diffraction imaging (CDI) [36], which is based on recording the light
scattered off a particle. In general, such a _diffraction pattern_ reflects
the scattering behavior of the particle, i.e., it depends on the distribution
of atoms and their response to the wavelength of the incident light.
Therefore, a change in scattering strength, e.g., because of a modification of
the particle’s density or its electron distribution, leads to a change of the
intensity distribution in the diffraction pattern, which is the basis for our
method. The intense femtosecond FEL light pulses enable taking a snapshot of
the particle in time, thus allowing tracking of the evolution of the particle
in a pump-probe measurement scheme. This technique has been recently used to
study light induced dynamics in a variety of systems, e.g., structural changes
in rare gas clusters [37, 38, 39, 40], surface melting of metal clusters [41],
plasma dynamics in SiO2 nanospheres [42], and anisotropic evaporation of
helium nanodroplets [18]. Here, we present a scenario where the energy is
coupled into the system via the dopants at localized positions. The agreement
between outstanding features in our recorded diffraction patterns and our
qualitative model of randomly distributed and ordered voids in the droplets
gives a strong indication that the dynamics is not homogeneous but starts at
several distinct sites in the droplets.

##  2 Methods

The investigation is based on the analysis of wide-angle scattering images
recorded at the FERMI FEL in a similar configuration to that previously
described [16]. The experimental setup is described in section 2.1. In section
2.2, the procedure to simulate wide-angle diffraction patterns in order to
retrieve information on the droplet dynamics from the recorded scattering
images is presented.

###  2.1 Pump-probe imaging setup

The experiment was carried out in a pump-probe measurement scheme at the low
density matter (LDM) end-station [43] of the FERMI FEL. The helium
nanodroplets are first irradiated with an NIR pulse (wavelength
$\lambda_{\mathrm{NIR}}=780~{}\mathrm{nm}$, duration $90~{}\mathrm{fs}$) to
trigger dynamics (_pump_) and subsequently, after a variable time delay of up
to $800~{}\mathrm{ps}$, imaged using a $90~{}\mathrm{fs}$ long XUV pulse
(_probe_). The NIR laser is focused to a $100~{}\mathrm{\mu m}$ spot
($1/\rme^{2}$ diameter) resulting in a power density of
$I_{\mathrm{NIR}}=9\times 10^{13}~{}\mathrm{W}\mathrm{cm}^{-2}$. The FEL is
tuned to either non-resonant or resonant photon energies
($E_{\mathrm{ph}}=19.4~{}\mathrm{eV}$ or $21.5~{}\mathrm{eV}$, respectively)
delivering power densities exceeding $I_{\mathrm{FEL}}=3\times
10^{14}~{}\mathrm{W}\mathrm{cm}^{-2}$ at spot sizes of $9\times
13~{}\mathrm{\mu m}^{2}$ (FWHM) and smaller. Please note that the focus size
of the NIR laser is chosen to be larger than that of the FEL to ensure that
each droplet is hit by the NIR laser before being imaged with the FEL. The XUV
light scattered off a helium droplet is collected up to a maximum scattering
angle $\theta_{\mathrm{max}}=30$° by a detector consisting of a circular
microchannel plate (MCP) stacked onto a phosphor screen [7] located
$65~{}\mathrm{mm}$ downstream of the interaction region. Pictures of the
amplified scattering images on the phosphor screen are taken via a 45° mirror
using a sCMOS Andor Neo 5.5 camera. The MCP, phosphor screen, and mirror have
centered holes to let the FEL and NIR beams pass through.

The helium droplets are produced by supersonic expansion into vacuum of helium
through a $100~{}\mathrm{\mu m}$ trumpet-shaped nozzle with half opening angle
of 20° at a stagnation pressure $p_{0}=80~{}\mathrm{bar}$ and temperature
$T_{0}=5.4~{}\mathrm{K}$, yielding a measured droplet velocity of
$320~{}\mathrm{m}\mathrm{s}^{-1}$ [44] and a mean droplet radius of $\langle
R\rangle=400~{}\mathrm{nm}$ (corresponding to a mean droplet size of $\langle
N\rangle=6\times 10^{9}$ atoms per droplet) as determined from scattering
images of approximately spherical droplets [44]. The droplets get doped with
xenon by traversing a $35.3~{}\mathrm{mm}$ long gas cell, successively
capturing $\sim 1.6\times 10^{6}$ xenon atoms per droplet (0.3 ‰) [45, 23].
The kinetic energy deposited in the droplet by each captured atom leads to the
evaporation of roughly 500 helium atoms [45, 46]. Hence, the average droplet
radius can be calculated to shrink to about $\langle
R^{\prime}\rangle=380~{}\mathrm{nm}$ [47], which matches well the measured
average radius of doped droplets. Overall, the dopant atoms constitute only a
small fraction of the total count of droplet atoms. Thus, for the wavelengths
used in this experiment we can assume the recorded diffraction patterns are
dominated by light scattered off the helium atoms.

In the analysis presented here, $26\,390$ images exhibiting meaningful
scattering signal have been included. The visibility of the scattering signal
has been improved in post processing of the data by subtracting the straylight
background. In addition, the images have been corrected for the uneven
detector sensitivity (that is, e.g., due to the 8° bias angle of the MCP [48])
and an angle-dependent intensity correction ($\propto\cos^{-3}\theta$) has
been applied that accounts for the flat shape of the detection locus [49]. The
thus corrected patterns are made available via the CXI Data Bank [50] under
the identifier (ID) 208 [51].

###  2.2 Wide-angle scattering simulations

In order to retrieve information on the droplet density distribution, the
diffraction patterns of individual droplets are analyzed. In the case of
_small-angle_ scattering ($\theta\lesssim 5$°) the droplet’s electron density
projection can be reconstructed from the diffraction pattern using a phase
retrieval algorithm which is commonly utilized in CDI experiments and has
already been successfully applied to doped helium nanodroplets [25, 26]. In
the case of _wide-angle_ scattering, however, multiple projection planes
contribute to the diffraction pattern rendering the reconstruction of a single
projected density using algorithms based on inverse 2D Fourier transform
impossible. For the same reason, wide-angle scattering images contain valuable
information on the three-dimensional droplet shape and orientation, that can
be retrieved by matching simulated diffraction patterns to the experimental
data, as has been shown for faceted silver clusters [52]. This _forward-
fitting_ technique has further been employed to investigate the shapes of
spinning helium nanodroplets [53, 54, 16] and is adapted here to address
density fluctuations inside the droplets via systematic simulations. The
choice of employing systematic simulations instead of directly deriving the
shape information from the diffraction patterns was enforced by the firm
constraints of the data set. These are (i) the large number of strongly
varying complex patterns, impossible to manually analyze in an appropriate
manner to generate reliable statistics of the abundances, (ii) the nonlinear
detector response together with the wide-angle detection which prevent object
reconstruction by iterative phase retrieval, and (iii) the large range of
complicated features from inhomogeneous densities in the evolved droplets that
could not be captured by generalized model shapes, which are needed for direct
forward fitting.

For calculating the wide-angle scattering patterns, a fast algorithm based on
the multi slice Fourier transform (MSFT) [55] approach was used. While its
approximations neglect multiple scattering events, e.g., backscattering of the
propagating electromagnetic field, material properties can be approximated via
effective optical parameters even for photon energies close to the resonance
[44]. These parameters have been determined by matching scattering simulations
to diffraction patterns of spherical helium nanodroplets and comparing the
radial intensity distribution to simulations based on Mie theory [56] for a
variety of droplet sizes and photon energies ranging from $19.0~{}\mathrm{eV}$
to $23.7~{}\mathrm{eV}$ [44].

##  3 Results

In the following, the measured scattering images are described and discussed.
First, the emergence of _dynamic features_ in the patterns is exemplified,
ı.e., of features that are not observed in static (XUV only) data. In section
3.1, the temporal evolution of the patterns is investigated, while in section
3.2 different classes of patterns are identified based on their salient
features.

Figure 1: Emergence of dynamic features in diffraction patterns of xenon-doped
helium nanodroplets. (a)-(c) For a time delay $\Delta t\leq 3~{}\mathrm{ps}$
the scattering images resemble those of pristine droplets described before
[16], i.e., the dopants do not lead per se to visible changes in the patterns.
(d)-(f) For longer delays, intensity fluctuations become apparent in the
images that are linked to the dynamics in the droplets and therefore referred
to as _dynamic features_.

Figure 1 shows a small selection of scattering images of xenon-doped helium
nanodroplets at short time delay $\Delta t$ between pump and probe pulse. It
should be noted that much more pronounced changes of the scattering occur at
longer delays which will be discussed later (cf. figures 3 and 4). In the
early stages of the dynamics, i.e., $\Delta t\leq 3\ \mathrm{ps}$, see figures
1(a)-(c), the images exhibit no visible difference from those of pristine
droplets [16]: In contrast to data taken with hard x-rays [26] the dopants do
not lead to a change of the intensity distribution. Presumably, the nanometer-
sized structures of dopant aggregates are not resolvable at the wavelength of
the incoming XUV pulse. However, as shown in figures 1(d)-(f), intensity
fluctuations along the diffraction rings become apparent for longer delays
$\Delta t>3\ \mathrm{ps}$. We assume these fluctuations are linked to the
dynamics in the droplets and consequently refer to these data as images
exhibiting _dynamic features_. Details on the determination of the images
showing dynamic features and an overview of the whole data set are given in A.
Overall, for short time delays and only slight intensity fluctuations, the
droplet shapes can still be inferred from the known characteristic patterns
[16] and classified as spherical [figures 1(a),(d)], ellipsoidal [figures
1(b),(e)], and pill-shaped [figures 1(c),(f)]. Nonetheless, the longer the
delay the more complicated it is to identify droplet shapes, as the
diffraction patterns get increasingly distorted (cf. figures 3 and 4).

###  3.1 Temporal evolution of the patterns

Figure 2: Temporal evolution of the fraction of scattering images exhibiting
dynamic features. After a steep increase within $100~{}\mathrm{ps}$ to
$150~{}\mathrm{ps}$ after NIR irradiation the dynamic fraction saturates at
about 60% to 80%, i.e., the dynamics is not ignited in every droplet. From the
inset it can be seen that for short delays ($\Delta t<3~{}\mathrm{ps}$) no
dynamics are observed in the patterns. Further, the rise of the dynamic
fraction is slower for the resonant photon energy
($E_{\mathrm{ph}}=21.5~{}\mathrm{eV}$), cf. the data points at 20 ps. The
dashed lines are limited growth functions as a guide to the eye.

In figure 2, the manually determined fraction of images exhibiting dynamic
features is shown versus the time delay $\Delta t$ for the non-resonant
($E_{\mathrm{ph}}=19.4\ \mathrm{eV}$) and resonant ($E_{\mathrm{ph}}=21.5\
\mathrm{eV}$) photon energy of the XUV pulse. The reproducibility of the
manually performed assignment of the dynamic fraction was tested (see A) to be
better than 90%. As a guide to the eye, limited growth functions are shown as
dashed lines. Three main observations can be made: (i) There is a steep
increase of the dynamic fraction, that is faster for the non-resonant
wavelength, (ii) the dynamic fraction saturates to less than 100% at long
delays ($\Delta t>200\ \mathrm{ps}$) indicating that not all images exhibit
dynamic features, and (iii) the saturation percentage is higher for the
resonant wavelength.

We attribute the first point to the wavelength-dependent influence of the
refractive index on the scattering. In the resonant case
($E_{\mathrm{ph}}=21.5\ \mathrm{eV}$) the absorption is high, hence, mostly
the front surface of the droplet contributes to the scattering and density
changes inside the droplet are not visible in the diffraction pattern. When
the dynamics initiated by the NIR pulse starts _inside_ the droplet (which we
expect to be the case for xenon dopants) and propagates towards the surface,
it can be observed earlier in the diffraction patterns recorded at the non-
resonant photon energy ($E_{\mathrm{ph}}=19.4\ \mathrm{eV}$). The second and
third observations, however, are most probably related to an experimental
artifact: We assume that an imperfect overlap of the pulses leads to a
situation where the dynamics is obviously not initiated in every droplet, even
though a larger focal spot was chosen for the pump (NIR) than for the probe
(XUV) pulse. Further, we identify the third observation – a slightly lower
asymptotic value for the non-resonant photon energy – as a decreasing overlap
between the two pulses over time, as even lower values were observed for
subsequent delay scans that were taken several hours apart (not shown).

Figure 3: The brightest images of individual xenon-doped helium nanodroplets
for short ($\sim$10 ps) and long ($\sim$800 ps) delays $\Delta t$ after
irradiation with the NIR laser pulse. Although the excitation process is the
same, the diffraction patterns look very different for the (a) non-resonant
($E_{\mathrm{ph}}=19.4~{}\mathrm{eV}$) and (b) resonant
($E_{\mathrm{ph}}=21.5~{}\mathrm{eV}$) photon energy. For example, the
intensity fluctuations at short delays are much less pronounced for resonant
scattering.

The dynamic features in the patterns become, as expected, more pronounced with
longer time delay. As well, the intensity not only fluctuates along the
diffraction rings, but the complexity of the intensity distribution increases.
It should be noted that many patterns exhibit strong directionality, which is
completely different from what was observed in earlier pump-probe experiments
on pure rare gas clusters [37, 38], thus indicating multiple scattering
centers at specific sites rather than surface smoothing or homogeneous
fragmentation of the particle. The development towards long time scales is
exemplified in figure 3, where the four brightest scattering images are shown
for short ($\sim$10 ps) and long ($\sim$800 ps) delays at non-resonant [figure
3(a)] and resonant [figure 3(b)] photon energy. It can be seen that the
intensity fluctuations at short delays are more pronounced for non-resonant
scattering ($E_{\mathrm{ph}}=19.4\ \mathrm{eV}$), where the droplets are
almost transparent and information on density fluctuations inside the droplets
is encoded in the diffraction pattern. For the resonant case
($E_{\mathrm{ph}}=21.5\ \mathrm{eV}$), less pronounced fluctuations presumably
reflect changes on the droplet surface that occur later in time. Note that
this observation is in line with the slower rise of the respective curve in
figure 2.

The images shown in figure 3 already exhibit complex features, especially at
long delays. However, this situation becomes even more complicated when not
only the brightest images are examined but the whole data set is analyzed.
While the overall trend – the longer the delay, the more distorted the
diffraction pattern – remains the same, the degree of complexity in the images
at a given delay varies considerably. In particular, no definite evolution in
time is discernible when individual patterns are compared. Two facts can be
considered to explain this observation: First, the scattering images reflect
individual droplets of different size, orientation, and shape, leading to a
huge variety of patterns. Second, as the position of the droplets with respect
to the NIR and the FEL focus varies from shot to shot, both the dynamics
triggered in the droplet and the intensity of the recorded scattering signal
are affected. Finally, a decrease of the spatial overlap, e.g., because of a
relative drift of the two beams, can further weaken the observable dynamics in
the pattern. In order to address the complexity of the data set, we therefore
further analyze the data by identifying classes of patterns with similar
characteristic features and by discussing their origins.

###  3.2 Pattern classification based on characteristic features

Figure 4: Classification of the dynamic data. Based on the characteristic
features in the diffraction patterns the images are manually grouped into six
categories: (a) Speckle patterns, (b) “spider” patterns, (c) polygon patterns,
(d) cross patterns, (e) mirror symmetric patterns, and (f) diverse patterns.

In order to investigate the shapes of individual nanoparticles from
diffraction patterns, it has previously proven helpful to group the scattering
images into a few classes based on their characteristic features [57, 58, 52,
16]. Here, we choose a similar approach to retrieve information about the
dynamics in the droplets. This is based on the assumption that similar changes
of the droplet density will lead to similar changes in the scattering images.
We identify patterns whose characteristic features stand out on the data set
and define classes for patterns exhibiting similar features. Figure 4
illustrates the variety as well as the complexity of the patterns in the data
set. A selection of classes is shown from which the first three will be
addressed in more detail in section 4. The first class, named _speckle
patterns_ , comprises patterns exhibiting speckles of different sizes and is
shown in figure 4(a). The second class, shown in figure 4(b), is called
_“spider” patterns_ , as streaks of higher intensity stick out of the pattern
like pairs of “legs” on opposite sides or, in a more asymmetric version, on
one side of the pattern. The third class consists of _polygon patterns_ ,
i.e., images exhibiting bright spots that define the corners of a polygon, see
figure 4(c). Two more classes are shown: Images exhibiting cross-like
structures in figure 4(d) and patterns with mirror symmetry, shown in figure
4(e). Intriguingly, the latter induce the perception of shapes like, e.g., a
butterfly, the face of a seal, or the greek letter $\Phi$, a phenomenon linked
to human pattern recognition generally known as _pareidolia_ that might be in
this case connected to the fractal characteristics of the patterns [59]. In
addition, most patterns do not fit into one of the presented categories,
examples are shown in figure 4(f). It is important to note that despite this,
the feature-based classification of the patterns makes it possible to develop
an idea of the underlying structures that lead to similar characteristic
scattering images, as we will exemplify in the following.

##  4 Modeling

In order to deduce possible fluctuations in the droplets from the scattering
images, we propose simple model shapes with fluctuating density that we use to
simulate wide-angle diffraction patterns. The goal of the simulations is to
qualitatively reproduce the features observed in the recorded images. To
minimize the influence of absorption effects, we deliberately constrain our
analysis to the non-resonant photon energy ($E_{\mathrm{ph}}=19.4\
\mathrm{eV}$).

As most of the helium nanodroplets exhibit spherical shapes (about 93%, see
[16]), we choose a sphere as the basis of our model. We mimic fluctuations of
the (optical) density in the droplet by introducing voids in the model shape
in the form of small spheres that can be placed either randomly or at specific
sites, e.g., along defined lines. This is motivated by the assumption that the
dynamics in the droplets is triggered by the NIR laser field at the positions
of the dopants, which presumably aggregate at multiple sites [24] or along
vortex lines [15]. The diameter of the spherical voids is randomly changed by
up to $\pm 20\%$ in order to avoid the occurrence of additional sharp features
in the simulated diffraction pattern. It should be noted that the chosen
representation of density fluctuations is not meant to imply any quantitative
information on their optical properties that are, however, difficult if not
impossible to determine. Nevertheless, also a qualitative comparison of the
calculated intensity distribution and the characteristic features in the
measured diffraction patterns enables tracing of structural changes occurring
in the droplets.

In particular, we aim at a more detailed analysis of the structures leading to
diffraction patterns in the speckle, “spider”, and polygon classes [cf.
figures 4(a)-(c)]. While the intensity fluctuations in the speckle patterns
[figure 4(a)] seem to be randomly distributed, the “spider” and polygon
patterns [figures 4(b),(c)] exhibit pronounced directionality. We assume a
connection between the fluctuations in the droplets and the intensity
distribution in the diffraction patterns and therefore treat the former case,
randomly distributed fluctuations, in section 4.1 and the latter case,
structured fluctuations, in section 4.2.

###  4.1 Randomly distributed fluctuations (speckle patterns)

Figure 5: Speckle patterns indicating randomly distributed density
fluctuations. In a simple model, we represent such fluctuations by voids or
“bubbles” randomly distributed in the droplet. As can be seen from the 3D
rendering of the model shape and its density projection, the number and size
of the “bubbles” is varied from (a) to (c). The simulated diffraction patterns
qualitatively reproduce the experimental data.

As has been discussed in section 3, intensity fluctuations in the scattering
images start to occur at delays $\Delta t>3\ \mathrm{ps}$ along the
diffraction rings (cf. figure 1): From hardly visible fluctuations to slight
variations of the intensity and a complete breakup of the rings, well-defined
speckles emerge that are randomly distributed in the patterns. Figure 5 shows
experimental data of speckle patterns with increasingly pronounced
fluctuations from (a) to (c). Based on the idea of multicenter dopant
aggregation in helium nanodroplets [24], we propose a sphere where bubble-like
voids are introduced as a simple model representing random fluctuations in a
droplet. The number and size of the “bubbles” can be varied while their
position in the droplet is chosen randomly. A 3D rendering of the model shape
and its density projection along the FEL axis are also depicted in figure 5.
Further, the corresponding simulated scattering image using the MSFT algorithm
as described in section 2.2 is shown. For a few and small “bubbles” [figure
5(a)], the diffraction rings are only slightly perturbed and the intensity
fluctuations are more pronounced at larger scattering angles (i.e., towards
the detector edge). When the number of “bubbles” is increased [figure 5(b)],
the diffraction rings remain intact only at small scattering angles while
towards the detector edge the rings get increasingly disrupted and speckles
form. In the case of larger “bubbles” [figure 5(c)] the speckle size increases
and the diffraction rings almost completely vanish, even for small scattering
angles. It can be seen that the calculated diffraction patterns qualitatively
reproduce the experimental data. This outcome indicates that the
irregularities are randomly distributed in the droplet.

###  4.2 Structured fluctuations (“spider” and polygon patterns)

Figure 6: “Spider” patterns indicating density fluctuations at specific sites
on the droplet surface. We model such fluctuations by placing multiple,
overlapping bubbles close to the droplet surface thus forming two dimples on
opposite sides. (a) When the axis connecting the dimples (dashed line) is
approximately perpendicular to the FEL axis (pink arrow), both dimples
contribute to the scattering leading to spider-like legs on two sides of the
pattern (dotted lines). (b) When the axis connecting the dimples is
considerably tilted, mostly the dimple closer to the light source contributes
to the scattering. Consequently, the spider-like legs are only visible to one
side of the diffraction pattern (dotted line).

In the case of scattering images exhibiting intensity fluctuations with strong
directionality, the diffraction patterns point to some underlying structure
governing the density fluctuations in the droplets. For example, in superfluid
helium nanodroplets, quantum vortices can dictate the arrangement of dopant
structures [15]. In the following, a model is presented that is able to
reproduce the observed scattering images of the “spider” and the polygon
classes [cf. figures 4(b),(c)].

Two examples of “spider” patterns are shown in figure 6. The peculiar feature
of these images is that they exhibit several pairs of streaks (reminiscent of
spider-like “legs”) with increased intensity towards larger scattering angles.
In addition to the experimental data the proposed model shape and its density
projection as well as the corresponding MSFT simulation are shown. The model
is constructed by placing several overlapping bubbles on or close to the
surface of a sphere, which leads to a spherical shape with dimples on opposite
sides. When the axis connecting these dimples is approximately perpendicular
to the FEL axis, as is shown in figure 6(a), the resulting diffraction pattern
exhibits the characteristic, spider-like legs (see the dotted lines in the
simulation). Figure 6(b) illustrates the case when the axis connecting the
dimples is considerably tilted with respect to the FEL axis. In consequence,
mostly the dimple closer to the light source contributes to the scattering,
which we attribute to absorption effects along the beam path, and the spider-
like legs occur only to one side of the pattern. The simulations qualitatively
reproduce the experimental data. Hence, these scattering images might be
caused by a single or a few irregularities at or close to the droplet surface.

Figure 7: Polygon patterns indicating a regular arrangement of the density
fluctuations. The patterns can be reproduced when the dimples are placed in a
polygon configuration while the axes connecting the dimples (dashed lines) are
oriented parallel to the FEL axis. The calculated diffraction patterns for (a)
triangular, (b) pentagonal, and (c) heptagonal configurations qualitatively
reproduce the experimental data.

Another conspicuous image type is that of polygon patterns, where bright spots
can be seen that are arranged like the corners of a polygon. Figure 7 shows
examples for (a) triangular, (b) pentagonal, and (c) heptagonal
configurations. We use the same model as before but with multiple dimples on
the droplet surface in a polygonal arrangement. The experimental data are
qualitatively reproduced by the simulations when the axes connecting the
dimples are approximately parallel to the FEL axis. While the positions of the
dimples are difficult to identify in the 3D model representation, they are
clearly visible in the density projection. Overall, dimples at specific sites
on the droplet surface can explain the observed diffraction patterns. While
the “spider” patterns point to two dimples on opposite sides of the droplet,
the polygon patterns imply a regular arrangement of multiple dimples. A
possible origin for these structures can be a grid-like distribution of the
dopants in the droplets, as will be discussed in the following section.

##  5 Discussion

Our observations of dynamics in xenon-doped helium nanodroplets are very
different from what has been previously seen for pure clusters. In contrast to
pristine systems, where nanoplasma dynamics linked to the NIR laser field have
been reported to lead to surface melting [37] or anisotropic evaporation [18],
the diffraction patterns of the heteronuclear system investigated here exhibit
complex features that point to fluctuations in the droplets, initiated at the
dopant positions. A huge variety of patterns is observed making it difficult
to assign their features to possible droplet configurations and their specific
temporal evolution. Nevertheless, we identified characteristic patterns that
suggest a random or a structured distribution of the dopants in the droplets.
These observations lead to three directions we want to further explore in the
following discussion: (i) What is the underlying process of the fluctuations?
(ii) Do the fluctuations occur in the volume or at the surface of the
droplets? (iii) How do the fluctuations evolve over time?

Up to now, theoretical studies of the interaction of intense NIR pulses with
doped helium nanodroplets have addressed the nanoplasma dynamics that evolve
in the droplets within a few hundreds of femtoseconds after irradiation [60,
19, 61, 20, 62]. The dynamics we observed in the diffraction patterns are on a
much longer timescale, which might be a direct result of the lower NIR power
density in our experiment (although the parameters – fraction of dopants in
the droplet and power density – are comparable to those given in [19]). The
scattering images remain unchanged for at least 3 ps, which is already a long
time compared to the calculated nanoplasma dynamics, and images exhibiting
bright scattering signal could be recorded up to 800 ps. On this timescale we
would expect that to a large extent recombination processes have already taken
place while the particle has fragmented [38]. Therefore, it is questionable
whether we directly observe the forming nanoplasma in the droplets, i.e.,
fluctuations of electronic origin. Another possibility is that fluctuations of
the droplet density occur in the form of a geometric displacement of the
droplet atoms, e.g., because of the formation of gas bubbles [28, 29]. In the
case of multiple dopant clusters randomly distributed in the droplet, this
could lead to the growth of multiple gas bubbles that become visible in the
diffraction patterns as speckles after a certain time, when they are large
enough to be detected. Note that the observed speckle patterns could also be
caused by strong fluctuations on the droplet surface. However, xenon dopants
are immersed in the droplets and do not remain on the surface [23]; further,
surface irregularities large enough to cause distortions of the diffraction
rings have not been observed for pure droplets [18]. Therefore, we assume the
fluctuations take place inside the droplets. In addition, the fact that the
dynamics in this experiment is observed earlier at the non-resonant photon
energy, where more information from the droplet interior contributes to the
scattering, could be a hint that fluctuations indeed start inside the droplet,
not on the surface. On the other hand, the “spider” patterns clearly indicate
that at least in some cases the fluctuations happen at or close to the
surface. While this could be in principle caused by bubbles propagating
towards the surface (that could eventually get ejected from the droplet),
there is no clear temporal evolution from patterns reflecting irregularities
inside the droplet to patterns showing changes on the droplet surface, i.e.,
speckle and “spider” patterns occur at the same time delay.

We therefore discuss another mechanism that has been described before: The
ejection of dopants from a helium droplet when they are ionized [31, 32]. In
this case, dopant atoms that aggregated close to the droplet surface probably
get ejected earlier and some helium atoms might be dragged away by the
dopants, thus leading to the formation of dimples on the surface. Furthermore,
dopants arranged along an array of vortices might therefore lead to a regular
arrangement of such fluctuations. In particular, the strong resemblance of the
polygon patterns observed in this experiment to previously reported
diffraction patterns of superfluid helium nanodroplets hosting quantized
vortices [26] suggests a connection between the vortex array and the density
fluctuations.

Next, we like to make some comments on the challenges of our analysis and how
to overcome them. The three guiding questions of our discussion, (i) to
identify an underlying process, (ii) to determine the locations of the
fluctuations, and (iii) to develop a picture of the evolving structures over
time, can only be partially answered by the analysis presented here:
Certainly, in order to substantiate our initial ideas on the dynamic processes
in the droplets and ultimately answer the questions raised by our analysis,
theoretical studies addressing both nanoplasma dynamics and structural
fluctuations are needed, especially for large droplets hosting multiple dopant
clusters or dopant structures that formed along vortex lines. Our results
underline that such calculations should span a time scale of several hundred
picoseconds. The temporal evolution of the fluctuations is even more difficult
to analyze due to the large variety of diffraction patterns, the lack of an
obvious time ordering of the observed patterns, and the extensive size of the
data set. Our analysis is focused on a small subset of distinctive patterns to
develop first ideas on the processes following irradiation with intense light
pulses. In order to trace the evolution of the dynamics, however, one needs to
identify patterns that are characteristic for a specific droplet configuration
at each delay. This is a very challenging and time consuming task that might
not even be possible to perform manually. Instead, it could be fruitful to
analyze the whole data set by using an automated approach for pattern
recognition, e.g., a neural network, as has been previously used for
classification of static diffraction patterns [63]. In particular, developing
an unsupervised network that can identify classes of characteristic patterns
for specific delays as an unbiased assistant to the human researcher would be
an interesting prospect for future work. In combination with an automated
forward-fitting routine enabling a quantitative 3D shape retrieval, e.g.,
based on a set of orthogonal functions and an efficient and fast MSFT
algorithm [55], the temporal evolution of the droplet shapes after NIR
irradiation could be finally revealed.

##  6 Conclusion

To summarize, we observed the dynamics in xenon-doped helium nanodroplets
after irradiation with an intense NIR pulse by analyzing wide-angle scattering
images recorded in a pump-probe experiment at the FERMI FEL. Our data show the
dynamics is linked to the xenon structures that formed in the droplets and
further corroborate the idea that the droplets do not expand homogeneously. As
the time scale of the observed dynamics is remarkably long (up to 800 ps), we
believe its origin to be rather geometric than electronic.

We analyzed the data by identifying scattering images that show distinctive
intensity distributions and grouping them into different classes based on
characteristic features in the diffraction patterns. In particular, we
associate scattering images exhibiting speckle patterns or “spider” and
polygon patterns to fluctuations that are randomly distributed in the droplet
or that occur at specific sites on or close to the droplet surface. We propose
that the fluctuations are connected to either multiple dopant clusters
randomly distributed in the droplet or dopants that agglomerate along
quantized vortices in array-like structures. In order to distinguish different
droplet configurations and trace their temporal evolution, it will be very
promising to establish a 3D-sensitive shape retrieval algorithm and an
automated classification of the scattering images, which are under
development.

Our results suggest that in addition to studies of nanoplasma evolution,
processes describing structural fluctuations such as bubble formation [28, 29]
or dopant ejection [31, 32] should be further investigated both experimentally
and theoretically, especially for the case of large doped helium nanodroplets.
As the considerable amount of energy deposited in the droplets by the NIR
pulse does not lead to their immediate destruction, additional dissipation
channels associated to the superfluid nature of the droplets should be
included in the models. In this context, a better understanding of the
underlying processes contributes to the prospect of using helium nanodroplets
as sacrificial layers in CDI experiments [64, 60, 65] or as an ultracold
environment for controlled nanostructure growth. Finally, using an NIR laser
pulse to magnify the structure of an embedded particle by creating gas bubbles
at the positions of its atoms, thus making it possible to analyze the
configuration of nanometer-sized objects even at relatively long (i.e., XUV)
wavelengths, is an interesting approach, especially in the context of
laboratory based experiments using high harmonic generation sources for single
particle imaging [54].

We would like to acknowledge excellent support by the FERMI staff during the
beam time. This work received financial support from the Deutsche
Forschungsgemeinschaft under the Grant No. MO 719/14-1, from the Leibniz-
Gemeinschaft under the Grant No. SAW/2017/MBI4, and from the Swiss National
Science Foundation under the Grant No. 200021E_193642 as well as via the NCCR
MUST program. F.S. acknowledges funding by Deutsche Forschungsgemeinschaft
under the Grant No. STI 125/19-2.

## Data availability statement

The complete data set is available at the Coherent X-Ray Imaging Data Bank
(CXIDB) [50] under the identifier (ID) 208: https://cxidb.org/id-208.html

##  Appendix A Determination of the dynamic fraction

In the following, we exemplify the classification of the data set to determine
the dynamic fraction (cf. figure 1), which is a measure for the number of
images that exhibit changes in the diffraction pattern that are due to the
prior irradiation with a strong NIR pulse. An overview of the data used for
this analysis is given in table 1. For each measurement run, the images
showing significant scattering signal (i.e., the “hits”) are filtered out by
simply thresholding the integrated intensity of the image. In a second step,
out of the total $26\,390$ hits, the images exhibiting dynamic features are
manually identified by the researcher. These are images that show features
that are not observed for the scattering images with only the XUV pulse
present (i.e., “static” data). This is further exemplified in figure 8. The
images shown in the first row, figure 8(a), show very smooth diffraction rings
that are not distorted, as it is known for static diffraction imaging data of
superfluid helium nanodroplets (cf. the scattering images presented in [16]).
In contrast, figure 8(b) shows examples of images exhibiting some kind of
distortion of the rings, hence, these images are labeled “dynamic”. In order
to illustrate the confidence of the classification, figure 8(c) shows images
exhibiting very subtle changes in the diffraction. They cannot be easily
assigned to one of the two classes, as it remains unclear whether the
distortion of the rings is caused by dynamic effects in the droplets or by,
e.g., inhomogeneities of the detector sensitivity. To give an estimate of the
accuracy of our classification, we additionally labeled a set of $1\,000$
randomly chosen scattering images as “static” or “dynamic” and compared the
results to the initial classification. For 93.6% of the images, the labels
agree with the initial assignment, while 4.8% of the images were initially
labeled wrong as “static” although they show clear dynamic features. For 1.6%
of the images the changes in the patterns are too subtle to decide whether
they show dynamic features or not.

Table 1: Data overview. For each measurement run, the images showing significant scattering signal (i.e., the “hits”) are identified. The images exhibiting dynamic features are manually determined by the researcher. For details, see text. Run No. |  $E_{\mathrm{ph}}$ (eV) |  $\Delta t$ (ps) | No. of shots | No. of hits | No. of images exhibiting
---|---|---|---|---|---
|  |  |  |  | dynamic features
153 | 21.5 | 200 | 12 000 | 1 897 | 1 455
155/156 | 21.5 | 500 | 12 000 | 1 704 | 1 254
157 | 21.5 | 805 | 12 000 | 1 555 | 1 189
162 | 21.5 | 294.3 | 11 300 | 1 391 | 1 120
163 | 21.5 | 0 | 12 000 | 1 458 | 0
164 | 21.5 | 9 | 12 000 | 1 601 | 327
165 | 21.5 | 1.5 | 12 000 | 1 677 | 0
166 | 21.5 | 5 | 12 000 | 1 636 | 27
167/168 | 21.5 | 3 | 12 000 | 1 602 | 0
169/170 | 21.5 | 20 | 12 000 | 1 676 | 376
175 | 19.4 | 0 | 10 000 | 1 429 | 0
176 | 19.4 | 8 | 10 000 | 1 773 | 550
177 | 19.4 | 50 | 10 000 | 1 667 | 1 020
178 | 19.4 | 800 | 10 000 | 1 105 | 735
179 | 19.4 | 20 | 10 000 | 1 597 | 813
180 | 19.4 | 150 | 10 000 | 1 407 | 993
181 | 19.4 | 400 | 10 000 | 1 215 | 824
Figure 8: Classification of static / dynamic images. (a) Scattering images
with no distortion of the diffraction rings are labelled “static”. (b) Images
exhibiting changes in the diffraction are labelled “dynamic”. (c) Examples for
images showing only subtle changes in the diffraction rings. Note that for the
images shown here, the correction for the uneven detector sensitivity (cf.
section 2.1) has not been applied.

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